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case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
| rw [← inv_mul_cancel_left (↑n) g, sup_comm] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ (↑n)⁻¹ * (↑n * g) ∈ N ⊔ normalizer (map (Subgroup.subtype N) ↑P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
| apply mul_mem_sup (N.inv_mem n.2) | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : n • MulAut.conjNormal g • P = P
⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
| rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
| refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_ | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
x : G
⊢ (MulEquiv.toMonoidHom (MulAut.conj (↑n * g))) x ∈
map (MulEquiv.toMonoidHom (MulAut.conj (↑n * g))) (map (Subgroup.subtype N) ↑P) ↔
↑n * g * x * (↑n * g)⁻¹ ∈ map (Subgroup.subtype N) ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
| rw [map_map, ← congr_arg (map N.subtype) hn, map_map] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
case intro
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
n : ↥N
hn : map ((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))) ↑P = ↑P
x : G
⊢ (MulEquiv.toMonoidHom (MulAut.conj (↑n * g))) x ∈
map (MonoidHom.comp (MulEquiv.toMonoidHom (MulAut.conj (↑n * g))) (Subgroup.subtype N)) ↑P ↔
↑n * g * x * (↑n * g)⁻¹ ∈
map
(MonoidHom.comp (Subgroup.subtype N)
((MulDistribMulAction.toMonoidEnd (MulAut ↥N) ↥N) (MulAut.conjNormal (↑n * g))))
↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
| rfl | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p G
hP : ↑P ≤ N
⊢ normalizer ↑P ⊔ N = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
| rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP] | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
| Mathlib.GroupTheory.Sylow.477_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
s : Subgroup G
t : Set (G ⧸ s)
⊢ Fintype.card ↑(mk ⁻¹' t) = Fintype.card ↥s * Fintype.card ↑t | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
| rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)] | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
| Mathlib.GroupTheory.Sylow.497_0.KwMUNfT2GXiDwTx | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : (n⁻¹ * x)⁻¹ * x ∈ H
⊢ x⁻¹ * n * x⁻¹⁻¹ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
| rw [mul_inv_rev, inv_inv] at this | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
| Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : x⁻¹ * n * x ∈ H
⊢ x⁻¹ * n * x⁻¹⁻¹ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
| convert this | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
| Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
case h.e'_4.h.e'_6
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ↑x ∈ MulAction.fixedPoints (↥H) (G ⧸ H)
ha : ∀ {y : G ⧸ H}, y ∈ orbit ↥H ↑x → y = ↑x
n : G
hn : n ∈ H
this : x⁻¹ * n * x ∈ H
⊢ x⁻¹⁻¹ = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
| rw [inv_inv] | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
| Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H
y✝ : G ⧸ H
y : G
hy : Quotient.mk'' y ∈ orbit ↥H ↑x
b : G
hb₁ : b ∈ H
hb₂✝ : (fun m => m • ↑x) { val := b, property := hb₁ } = Quotient.mk'' y
hb₂ : (b * x)⁻¹ * y ∈ H
⊢ b⁻¹ * (x * (y⁻¹ * x)⁻¹ * x⁻¹) ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
| rw [hx] at hb₂ | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
| Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H
y✝ : G ⧸ H
y : G
hy : Quotient.mk'' y ∈ orbit ↥H ↑x
b : G
hb₁ : b ∈ H
hb₂✝ : (fun m => m • ↑x) { val := b, property := hb₁ } = Quotient.mk'' y
hb₂ : x * ((b * x)⁻¹ * y) * x⁻¹ ∈ H
⊢ b⁻¹ * (x * (y⁻¹ * x)⁻¹ * x⁻¹) ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; | simpa [mul_inv_rev, mul_assoc] using hb₂ | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; | Mathlib.GroupTheory.Sylow.503_0.KwMUNfT2GXiDwTx | theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
⊢ ∀ (x y : Subtype ↑(normalizer H)), Setoid.r x y ↔ ↑x ≈ ↑y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
| intros | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
| Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ Setoid.r x✝ y✝ ↔ ↑x✝ ≈ ↑y✝ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
| dsimp only [instHasEquiv] | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
| Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ Setoid.r x✝ y✝ ↔ Setoid.r ↑x✝ ↑y✝ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
| rw [leftRel_apply (α := normalizer H), leftRel_apply] | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
| Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x✝ y✝ : Subtype ↑(normalizer H)
⊢ x✝⁻¹ * y✝ ∈ comap (Subgroup.subtype (normalizer H)) H ↔ (↑x✝)⁻¹ * ↑y✝ ∈ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
| rfl | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
| Mathlib.GroupTheory.Sylow.526_0.KwMUNfT2GXiDwTx | /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card (↥(normalizer H) ⧸ comap (Subgroup.subtype (normalizer H)) H) ≡ Fintype.card (G ⧸ H) [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
| rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)] | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
| Mathlib.GroupTheory.Sylow.540_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card ↑(MulAction.fixedPoints (↥H) (G ⧸ H)) ≡ Fintype.card (G ⧸ H) [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
| exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
| Mathlib.GroupTheory.Sylow.540_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
⊢ Fintype.card ↥(normalizer H) ≡ Fintype.card G [MOD p ^ (n + 1)] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
| have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
| Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
this : ↥(subgroupOf H (normalizer H)) ≃ ↥H
⊢ Fintype.card ↥(normalizer H) ≡ Fintype.card G [MOD p ^ (n + 1)] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
| rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
| Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
this : ↥(subgroupOf H (normalizer H)) ≃ ↥H
⊢ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) * p ^ n ≡ Fintype.card (G ⧸ H) * p ^ n [MOD p * p ^ n] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
| exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _ | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
| Mathlib.GroupTheory.Sylow.550_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
⊢ Fintype.card (G ⧸ H) * Fintype.card ↥H = s * p * Fintype.card ↥H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
| rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p] | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
| Mathlib.GroupTheory.Sylow.561_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ comap (Subgroup.subtype (normalizer H)) H) % p
⊢ Fintype.card (↥(normalizer H) ⧸ comap (Subgroup.subtype (normalizer H)) H) % p = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by | rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by | Mathlib.GroupTheory.Sylow.561_0.KwMUNfT2GXiDwTx | /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
⊢ Fintype.card (G ⧸ H) * Fintype.card ↥H = s * p * Fintype.card ↥H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
| rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
⊢ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by | rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
⊢ Fintype.card
↥(Subgroup.map (Subgroup.subtype (normalizer H)) (comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) =
p ^ (n + 1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
| show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1) | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
⊢ Fintype.card
↥(Subgroup.map (Subgroup.subtype (normalizer H)) (comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) =
p ^ (n + 1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
| suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2 | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
this : Fintype.card ↑(Subtype.val '' ↑(comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) = p ^ (n + 1)
⊢ Fintype.card
↥(Subgroup.map (Subgroup.subtype (normalizer H)) (comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) =
p ^ (n + 1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by | convert this using 2 | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by | Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
⊢ Fintype.card ↑(Subtype.val '' ↑(comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) = p ^ (n + 1) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
| rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
⊢ Fintype.card ↑↑(comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) =
Fintype.card (↥(subgroupOf H (normalizer H)) × ↥(zpowers x)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
| exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)) | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
⊢ H ≤ Subgroup.map (Subgroup.subtype (normalizer H)) (comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
| intro y hy | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
y : G
hy : y ∈ H
⊢ y ∈ Subgroup.map (Subgroup.subtype (normalizer H)) (comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
| simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
y : G
hy : y ∈ H
⊢ ∃ x_1, ↑x_1 ∈ zpowers x ∧ ↑x_1 = y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
| refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩ | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
y : G
hy : y ∈ H
⊢ (fun x_1 => x ^ x_1) 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
| dsimp only | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
y : G
hy : y ∈ H
⊢ x ^ 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
| rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
s : ℕ
hs : Fintype.card G = s * p ^ (n + 1)
hcard : Fintype.card (G ⧸ H) = s * p
hm : s * p % p = Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H)) % p
hm' : p ∣ Fintype.card (↥(normalizer H) ⧸ subgroupOf H (normalizer H))
x : ↥(normalizer H) ⧸ subgroupOf H (normalizer H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(subgroupOf H (normalizer H))
y : G
hy : y ∈ H
⊢ { val := y, property := (_ : y ∈ normalizer H) } ∈ subgroupOf H (normalizer H) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
| simpa using hy | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
| Mathlib.GroupTheory.Sylow.586_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
_wf : m - 1 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Fintype.card ↥K = p ^ (m - 1) ∧ H ≤ K
⊢ p ^ (m - 1 + 1) ∣ Fintype.card G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by | rwa [tsub_add_cancel_of_le h0m.nat_succ_le] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
_wf : m - 1 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Fintype.card ↥K = p ^ (m - 1) ∧ H ≤ K
hdvd' : p ^ (m - 1 + 1) ∣ Fintype.card G
K' : Subgroup G
hK' : Fintype.card ↥K' = p ^ (m - 1 + 1) ∧ K ≤ K'
⊢ Fintype.card ↥K' = p ^ m | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by | rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Fintype.card G
H : Subgroup G
hH : Fintype.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n = m
⊢ Fintype.card ↥H = p ^ m ∧ H ≤ H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by | simp [hH, hnm] | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by | Mathlib.GroupTheory.Sylow.627_0.KwMUNfT2GXiDwTx | /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p n : ℕ
inst✝ : Fact (Nat.Prime p)
hdvd : p ^ n ∣ Fintype.card G
⊢ 1 = p ^ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by | simp | /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by | Mathlib.GroupTheory.Sylow.648_0.KwMUNfT2GXiDwTx | /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
| have : Fact p.Prime := ⟨hp⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
| have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ Nat.card G ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by | linarith [Nat.one_le_pow n p hp.pos] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by | Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
| cases nonempty_fintype G | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
| obtain ⟨m, hm⟩ := h.exists_card_eq | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
⊢ ∃ H, Nat.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
| simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ Fintype.card G
⊢ ∃ H, Fintype.card ↥H = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
| refine exists_subgroup_card_pow_prime _ ?_ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ Fintype.card G
⊢ p ^ n ∣ Fintype.card G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
| rw [hm] at hn ⊢ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
val✝ : Fintype G
m : ℕ
hm : Fintype.card G = p ^ m
hn : p ^ n ≤ p ^ m
⊢ p ^ n ∣ p ^ m | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
| exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
| Mathlib.GroupTheory.Sylow.656_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
⊢ ∃ H' ≤ H, Nat.card ↥H' = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
| obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
| Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ ∃ H' ≤ H, Nat.card ↥H' = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
| refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
| Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ Nat.card ↥(Subgroup.map (Subgroup.subtype H) H') = p ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
| rw [← H'card] | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
| Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
⊢ Nat.card ↥(Subgroup.map (Subgroup.subtype H) H') = Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
| let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
| Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
H' : Subgroup ↥H
H'card : Nat.card ↥H' = p ^ n
e : ↥H' ≃* ↥(Subgroup.map (Subgroup.subtype H) H') :=
equivMapOfInjective H' (Subgroup.subtype H) (_ : Injective ⇑(Subgroup.subtype H))
⊢ Nat.card ↥(Subgroup.map (Subgroup.subtype H) H') = Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
| exact Nat.card_congr e.symm.toEquiv | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
| Mathlib.GroupTheory.Sylow.669_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
| obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
| Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
| obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk) | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
| Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' | Mathlib_GroupTheory_Sylow |
case intro.intro.intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
H' : Subgroup G
H'H : H' ≤ H
H'card : Nat.card ↥H' = p ^ m
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
| refine ⟨H', H'H, ?_⟩ | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
| Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' | Mathlib_GroupTheory_Sylow |
case intro.intro.intro.intro
G : Type u
α : Type v
β : Type w
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
m : ℕ
hmk : p ^ m ≤ k
hkm : k < p ^ (m + 1)
H' : Subgroup G
H'H : H' ≤ H
H'card : Nat.card ↥H' = p ^ m
⊢ Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
| simpa only [pow_succ, H'card] using And.intro hmk hkm | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
| Mathlib.GroupTheory.Sylow.679_0.KwMUNfT2GXiDwTx | /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
| rw [← pow_one p] at hdvd | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ^ 1 ∣ Fintype.card G
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
| have key := P.pow_dvd_card_of_pow_dvd_card hdvd | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ^ 1 ∣ Fintype.card G
key : p ^ 1 ∣ Fintype.card ↥↑P
⊢ p ∣ Fintype.card ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
| rwa [pow_one] at key | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
| Mathlib.GroupTheory.Sylow.696_0.KwMUNfT2GXiDwTx | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
⊢ ↑P ≠ ⊥ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
| refine' fun h => hp.out.not_dvd_one _ | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
h : ↑P = ⊥
⊢ p ∣ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
| have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Fintype.card G
h : ↑P = ⊥
key : p ∣ Fintype.card ↥↑P
⊢ p ∣ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
| rwa [h, card_bot] at key | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
| Mathlib.GroupTheory.Sylow.710_0.KwMUNfT2GXiDwTx | theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
⊢ Fintype.card ↥↑P = p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
| obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup' | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P = p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
| refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p)) | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P ∣ p ^ (Nat.factorization (Fintype.card G)) p | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
| rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq] | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Fintype.card ↥↑P = p ^ n
⊢ Fintype.card ↥↑P ∣ Fintype.card G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
| exact P.1.card_subgroup_dvd_card | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
| Mathlib.GroupTheory.Sylow.717_0.KwMUNfT2GXiDwTx | /-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Fintype G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : Fintype ↥H
card_eq : Fintype.card ↥H = p ^ (Nat.factorization (Fintype.card G)) p
⊢ ∀ {Q : Subgroup G}, IsPGroup p ↥Q → H ≤ Q → Q = H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
| obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
| Mathlib.GroupTheory.Sylow.727_0.KwMUNfT2GXiDwTx | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Fintype G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
H : Subgroup G
inst✝ : Fintype ↥H
card_eq : Fintype.card ↥H = p ^ (Nat.factorization (Fintype.card G)) p
P : Sylow p G
hHP : H ≤ ↑P
⊢ ∀ {Q : Subgroup G}, IsPGroup p ↥Q → H ≤ Q → Q = H | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
| exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3 | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
| Mathlib.GroupTheory.Sylow.727_0.KwMUNfT2GXiDwTx | /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
⊢ Unique (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
| refine { uniq := fun Q ↦ ?_ } | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
| obtain ⟨x, h1⟩ := exists_smul_eq G P Q | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x : G
h1 : x • P = Q
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
| obtain ⟨x, h2⟩ := exists_smul_eq G P default | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x✝ : G
h1 : x✝ • P = Q
x : G
h2 : x • P = default
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
| rw [Sylow.smul_eq_of_normal] at h1 h2 | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
case intro.intro
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
Q : Sylow p G
x✝ : G
h1 : P = Q
x : G
h2 : P = default
⊢ Q = default | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
| rw [← h1, ← h2] | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
| Mathlib.GroupTheory.Sylow.746_0.KwMUNfT2GXiDwTx | /-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
⊢ Characteristic ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
| haveI := Sylow.unique_of_normal P h | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
⊢ Characteristic ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
| rw [characteristic_iff_map_eq] | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
⊢ ∀ (ϕ : G ≃* G), Subgroup.map (MulEquiv.toMonoidHom ϕ) ↑P = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
| intro Φ | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ Subgroup.map (MulEquiv.toMonoidHom Φ) ↑P = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
| show (Φ • P).toSubgroup = P.toSubgroup | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ ↑(Φ • P) = ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
| congr | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
case e_self
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : Normal ↑P
this : Unique (Sylow p G)
Φ : G ≃* G
⊢ Φ • P = P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
| simp [eq_iff_true_of_subsingleton] | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
| Mathlib.GroupTheory.Sylow.760_0.KwMUNfT2GXiDwTx | theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hn : Normal (normalizer ↑P)
⊢ Normal ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
| rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem] | theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
| Mathlib.GroupTheory.Sylow.772_0.KwMUNfT2GXiDwTx | theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer (normalizer ↑P) = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
| have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer)) | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this :
Normal (normalizer ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))) →
Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer (normalizer ↑P)))
⊢ normalizer (normalizer ↑P) = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
| simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤
⊢ normalizer (normalizer ↑P) = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
| rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial] | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this : True → subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P)) = ⊤
⊢ Subgroup.map (Subgroup.subtype (normalizer (normalizer ↑P)))
(subgroupOf (normalizer ↑P) (normalizer (normalizer ↑P))) =
normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
| exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _))) | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
| Mathlib.GroupTheory.Sylow.777_0.KwMUNfT2GXiDwTx | @[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
| rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩) | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
| Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inl
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
heq : normalizer ↑P = ⊤
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· | exact heq | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· | haveI := hnc _ hK | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· | Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
| have hPK : ↑P ≤ K := le_trans le_normalizer hNK | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
| Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
hPK : ↑P ≤ K
⊢ normalizer ↑P = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
| refine' (hK.1 _).elim | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
| Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
case inr.intro.intro
G : Type u
α : Type v
β : Type w
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → Normal H
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
this : Normal K
hPK : ↑P ≤ K
⊢ K = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
| rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK] | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
| Mathlib.GroupTheory.Sylow.788_0.KwMUNfT2GXiDwTx | theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
⊢ ((p : { x // x ∈ (Fintype.card G).primeFactors }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
| set ps := (Fintype.card G).primeFactors | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
⊢ ((p : { x // x ∈ ps }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
| let P : ∀ p, Sylow p G := default | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
⊢ ((p : { x // x ∈ ps }) → (P : Sylow (↑p) G) → ↥↑P) ≃* G | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
| have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
have hne' : p₁ ≠ p₂ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂))
apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup' | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
⊢ _root_.Pairwise fun p₁ p₂ => ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑p₂ → Commute x y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
| rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val := p₂, property := hp₂ }
⊢ ∀ (x y : G), x ∈ P ↑{ val := p₁, property := hp₁ } → y ∈ P ↑{ val := p₂, property := hp₂ } → Commute x y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
| haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val := p₂, property := hp₂ }
hp₁' : Fact (Nat.Prime p₁)
⊢ ∀ (x y : G), x ∈ P ↑{ val := p₁, property := hp₁ } → y ∈ P ↑{ val := p₂, property := hp₂ } → Commute x y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
| haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂) | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Fintype G
hn : ∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal ↑P
ps : Finset ℕ := (Fintype.card G).primeFactors
P : (p : ℕ) → Sylow p G := default
p₁ : ℕ
hp₁ : p₁ ∈ ps
p₂ : ℕ
hp₂ : p₂ ∈ ps
hne : { val := p₁, property := hp₁ } ≠ { val := p₂, property := hp₂ }
hp₁' : Fact (Nat.Prime p₁)
hp₂' : Fact (Nat.Prime p₂)
⊢ ∀ (x y : G), x ∈ P ↑{ val := p₁, property := hp₁ } → y ∈ P ↑{ val := p₂, property := hp₂ } → Commute x y | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn
refine' fun x =>
(mem_map_iff_mem
(show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
(MulAut.conj (↑n * g)).injective)).symm.trans
_
rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
rfl
#align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : ↑P ≤ N) : (P : Subgroup G).normalizer ⊔ N = ⊤ := by
rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroupOf_map_subtype,
inf_of_le_left hP]
#align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
open BigOperators
universe u v w
variable {G : Type u} {α : Type v} {β : Type w} [Group G]
attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
#align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
namespace Sylow
theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
{x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
⟨fun hx =>
have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n ∈ H) =>
have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
show _ ∈ H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let ⟨⟨b, hb₁⟩, hb₂⟩ := hy
have hb₂ : (b * x)⁻¹ * y ∈ H := QuotientGroup.eq.1 hb₂
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hb₁)).1 <| by
rw [hx] at hb₂; simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
#align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
/-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
MulAction.fixedPoints H (G ⧸ H) ≃
normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G ⧸ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (α := normalizer H), leftRel_apply]
rfl)
#align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
/-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
mod `p` to the index of `H`. -/
theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
#align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
hH, pow_succ]
exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
#align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
index of `H` inside its normalizer. -/
theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ∣ card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm :
s * p % p =
card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) % p :=
hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
#align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
/-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) ∣ card (normalizer H) :=
Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
#align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
if `p ^ (n + 1)` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) ∣ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
∃ K : Subgroup G, Fintype.card K = p ^ (n + 1) ∧ H ≤ K :=
let ⟨s, hs⟩ := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G ⧸ H) = s * p :=
(mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
(by
rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
⟨Subgroup.map (normalizer H).subtype
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x)), by
show Fintype.card (Subgroup.map H.normalizer.subtype
(comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)
suffices Fintype.card (Subtype.val ''
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =
p ^ (n + 1)
by convert this using 2
rw [Set.card_image_of_injective
(Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
Subtype.val_injective,
pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
Fintype.card_prod]
exact @Fintype.card_congr _ _ (_) (_)
(preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
intro y hy
simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hy⟩
#align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
/-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
then `H` is contained in a subgroup of cardinality `p ^ m`
if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
∀ {n m : ℕ} [_hp : Fact p.Prime] (_hdvd : p ^ m ∣ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n ≤ m), ∃ K : Subgroup G, card K = p ^ m ∧ H ≤ K
| n, m => fun {hdvd H hH hnm} =>
(lt_or_eq_of_le hnm).elim
(fun hnm : n < m =>
have h0m : 0 < m := lt_of_le_of_lt n.zero_le hnm
have _wf : m - 1 < m := Nat.sub_lt h0m zero_lt_one
have hnm1 : n ≤ m - 1 := le_tsub_of_add_le_right hnm
let ⟨K, hK⟩ :=
@exists_subgroup_card_pow_prime_le _ _ n (m - 1) _
(Nat.pow_dvd_of_le_of_pow_dvd tsub_le_self hdvd) H hH hnm1
have hdvd' : p ^ (m - 1 + 1) ∣ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let ⟨K', hK'⟩ := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
#align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
/-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ card G) : ∃ K : Subgroup G, Fintype.card K = p ^ n :=
let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (card_bot.trans (by simp)) n.zero_le
⟨K, hK.1⟩
#align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
then there is a subgroup of cardinality `p ^ n`. -/
lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := ⟨hp⟩
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain ⟨m, hm⟩ := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn ⊢
exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
rw [← H'card]
let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
exact Nat.card_congr e.symm.toEquiv
/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine ⟨H', H'H, ?_⟩
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
(hp.1.coprime_pow_of_not_dvd
(not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
((index_mul_card P.1).symm ▸ hdvd)
#align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
#align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
/-- Sylow subgroups are Hall subgroups. -/
theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
#align sylow.card_coprime_index Sylow.card_coprime_index
theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ := by
refine' fun h => hp.out.not_dvd_one _
have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
rwa [h, card_bot] at key
#align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
/-- The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order. -/
theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain ⟨n, heq : card P = _⟩ := IsPGroup.iff_card.mp P.isPGroup'
refine' Nat.dvd_antisymm _ (P.pow_dvd_card_of_pow_dvd_card (Nat.ord_proj_dvd _ p))
rw [heq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← heq]
exact P.1.card_subgroup_dvd_card
#align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
where `n` is the multiplicity of `p` in the group order. -/
def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : Sylow p G
where
toSubgroup := H
isPGroup' := IsPGroup.of_card card_eq
is_maximal' := by
obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
exact SetLike.ext'
(Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
rfl
#align sylow.coe_of_card Sylow.coe_ofCard
/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
refine { uniq := fun Q ↦ ?_ }
obtain ⟨x, h1⟩ := exists_smul_eq G P Q
obtain ⟨x, h2⟩ := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [← h1, ← h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
#align sylow.characteristic_of_normal Sylow.characteristic_of_normal
end Pointwise
theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
#align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
@[simp]
theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [← normalizer_eq_top, Sylow.coe_subtype, ← subgroupOf_normalizer_eq le_normalizer, ←
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
← this trivial]
exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
#align sylow.normalizer_normalizer Sylow.normalizer_normalizer
theorem normal_of_all_max_subgroups_normal [Finite G]
(hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (↑P : Subgroup G).normalizer with (heq | ⟨K, hK, hNK⟩)
· exact heq
· haveI := hnc _ hK
have hPK : ↑P ≤ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
#align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
normalizer_eq_top.mp <|
normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
#align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
open BigOperators
/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
| have hne' : p₁ ≠ p₂ := by simpa using hne | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
set ps := (Fintype.card G).primeFactors
-- “The” Sylow subgroup for p
let P : ∀ p, Sylow p G := default
have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
| Mathlib.GroupTheory.Sylow.809_0.KwMUNfT2GXiDwTx | /-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
of these Sylow subgroups.
-/
noncomputable def directProductOfNormal [Fintype G]
(hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
(∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G | Mathlib_GroupTheory_Sylow |
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