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G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
β’ pβ β pβ | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
hne' : pβ β pβ
β’ β (x y : G), x β P β{ val := pβ, property := hpβ } β y β P β{ val := pβ, property := hpβ } β 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
| apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ)) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
hne' : pβ β pβ
β’ _root_.Disjoint β(P pβ) β(P pβ) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
| apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β (P : Sylow (βp) G) β β₯βP) β* 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
| refine' MulEquiv.trans (N := β p : ps, P p) _ _ | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β (P : Sylow (βp) G) β β₯βP) β* ((p : { x // x β ps }) β β₯(P βp))
case refine'_2
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β β₯(P βp)) β* 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
| show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β (P : Sylow (βp) G) β β₯P) β* ((p : { x // x β ps }) β β₯(P βp)) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
| apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _ | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ (j : { x // x β ps }) β ((P : Sylow (βj) G) β β₯P) β* β₯β(P βj) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
| rintro β¨p, hpβ© | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
p : β
hp : p β ps
β’ ((P : Sylow (β{ val := p, property := hp }) G) β β₯P) β* β₯β(P β{ val := p, property := hp }) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
| haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
p : β
hp : p β ps
hp' : Fact (Nat.Prime p)
β’ ((P : Sylow (β{ val := p, property := hp }) G) β β₯P) β* β₯β(P β{ val := p, property := hp }) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
| letI := unique_of_normal _ (hn (P p)) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_1.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
p : β
hp : p β ps
hp' : Fact (Nat.Prime p)
this : Unique (Sylow p G) := unique_of_normal (P p) (_ : Normal β(P p))
β’ ((P : Sylow (β{ val := p, property := hp }) G) β β₯P) β* β₯β(P β{ val := p, property := hp }) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
| apply MulEquiv.piUnique | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β β₯(P βp)) β* 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
| show (β p : ps, P p) β* G | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ ((p : { x // x β ps }) β β₯(P βp)) β* 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
| apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Bijective β(noncommPiCoprod hcomm) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
| apply (bijective_iff_injective_and_card _).mpr | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Injective β(noncommPiCoprod hcomm) β§ Fintype.card ((i : { x // x β ps }) β β₯β(P βi)) = 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
| constructor | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Injective β(noncommPiCoprod hcomm)
case refine'_2.right
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Fintype.card ((i : { x // x β ps }) β β₯β(P βi)) = 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
| show Injective _ | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Injective β(noncommPiCoprod hcomm) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· | apply Subgroup.injective_noncommPiCoprod_of_independent | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· | 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ CompleteLattice.Independent fun i => β(P βi) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
| apply independent_of_coprime_order hcomm | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ β (i j : { x // x β ps }), i β j β Nat.Coprime (Fintype.card β₯β(P βi)) (Fintype.card β₯β(P βj)) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
| rintro β¨pβ, hpββ© β¨pβ, hpββ© 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
β’ Nat.Coprime (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
| haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
β’ Nat.Coprime (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
| haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
β’ Nat.Coprime (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
| have hne' : pβ β pβ := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
β’ pβ β pβ | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.left.hind.mk.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
pβ : β
hpβ : pβ β ps
pβ : β
hpβ : pβ β ps
hne : { val := pβ, property := hpβ } β { val := pβ, property := hpβ }
hpβ' : Fact (Nat.Prime pβ)
hpβ' : Fact (Nat.Prime pβ)
hne' : pβ β pβ
β’ Nat.Coprime (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) (Fintype.card β₯β(P β{ val := pβ, property := hpβ })) | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
| apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.right
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Fintype.card ((i : { x // x β ps }) β β₯β(P βi)) = 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
| show card (β p : ps, P p) = card G | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case refine'_2.right
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ Fintype.card ((p : { x // x β ps }) β β₯(P βp)) = 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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· | calc
card (β p : ps, P p) = β p : ps, card (P p) := Fintype.card_pi
_ = β p : ps, p.1 ^ (card G).factorization p.1 := by
congr 1 with β¨p, hpβ©
exact @card_eq_multiplicity _ _ _ p β¨Nat.prime_of_mem_primeFactors hpβ© (P p)
_ = β p in ps, p ^ (card G).factorization p :=
(Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
_ = (card G).factorization.prod (Β· ^ Β·) := rfl
_ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· | 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
β’ β p : { x // x β ps }, Fintype.card β₯β(P βp) = β p : { x // x β ps }, βp ^ (Nat.factorization (Fintype.card G)) βp | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· calc
card (β p : ps, P p) = β p : ps, card (P p) := Fintype.card_pi
_ = β p : ps, p.1 ^ (card G).factorization p.1 := by
| congr 1 with β¨p, hpβ© | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· calc
card (β p : ps, P p) = β p : ps, card (P p) := Fintype.card_pi
_ = β p : ps, p.1 ^ (card G).factorization p.1 := 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
case e_f.h.mk
G : Type u
Ξ± : Type v
Ξ² : Type w
instβΒΉ : Group G
instβ : Fintype G
hn : β {p : β} [inst : Fact (Nat.Prime p)] (P : Sylow p G), Normal βP
ps : Finset β := (Fintype.card G).primeFactors
P : (p : β) β Sylow p G := default
hcomm : _root_.Pairwise fun pβ pβ => β (x y : G), x β P βpβ β y β P βpβ β Commute x y
p : β
hp : p β ps
β’ Fintype.card β₯β(P β{ val := p, property := hp }) =
β{ val := p, property := hp } ^ (Nat.factorization (Fintype.card G)) β{ val := p, property := hp } | /-
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 `nβ` be the number of Sylow `p`-subgroups of `G`, then `nβ` divides the index of the Sylow
`p`-subgroup, `nβ β‘ 1 [MOD p]`, and `nβ` 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 `pβΏ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pβΏ`.
* `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 : β) (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' : β {Q : Subgroup G}, IsPGroup p Q β toSubgroup β€ Q β 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) :=
β¨Sylow.toSubgroupβ©
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = βP :=
-- 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 β (P : Subgroup G) = Q :=
β¨congr_arg _, extβ©
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (β)
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 {Ξ± : Type*} [MulAction G Ξ±] : MulAction P Ξ± :=
inferInstanceAs (MulAction (P : Subgroup G) Ξ±)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (Ο : K β* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) : Sylow p K :=
{ P.1.comap Ο with
isPGroup' := P.2.comap_of_ker_isPGroup Ο hΟ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap Ο
rw [β P.3 (hQ.map Ο) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap Ο).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hΟ : IsPGroup p Ο.ker) (h : βP β€ Ο.range) :
(P.comapOfKerIsPGroup Ο hΟ h : Subgroup K) = Subgroup.comap Ο βP :=
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 (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) : Sylow p K :=
P.comapOfKerIsPGroup Ο (IsPGroup.ker_isPGroup_of_injective hΟ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hΟ : Function.Injective Ο) (h : βP β€ Ο.range) :
β(P.comapOfInjective Ο hΟ h) = Subgroup.comap Ο βP :=
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 : βP β€ 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 : βP β€ N) : β(P.subtype h) = subgroupOf (βP) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : βP β€ N} {hQ : βQ β€ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h β’
exact fun g => β¨fun hg => (h β¨g, hP hgβ©).mp hg, fun hg => (h β¨g, hQ hgβ©).mpr hgβ©
#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) : β Q : Sylow p G, P β€ Q :=
Exists.elim
(zorn_nonempty_partialOrderβ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
β¨{ carrier := β R : c, R
one_mem' := β¨Q, β¨β¨Q, hQβ©, rflβ©, Q.one_memβ©
inv_mem' := fun {g} β¨_, β¨R, rflβ©, hgβ© => β¨R, β¨R, rflβ©, R.1.inv_mem hgβ©
mul_mem' := fun {g} h β¨_, β¨R, rflβ©, hgβ© β¨_, β¨S, rflβ©, hhβ© =>
(hc2.total R.2 S.2).elim (fun T => β¨S, β¨S, rflβ©, S.1.mul_mem (T hg) hhβ©) fun T =>
β¨R, β¨R, rflβ©, R.1.mul_mem hg (T hh)β© },
fun β¨g, _, β¨S, rflβ©, hgβ© => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 β¨g, hgβ©)
rwa [Subtype.ext_iff, coe_pow] at hk β’, fun M hM g hg => β¨M, β¨β¨M, hMβ©, rflβ©, hgβ©β©)
P hP)
fun {Q} β¨hQ1, hQ2, hQ3β© => β¨β¨Q, hQ1, hQ3 _β©, hQ2β©
#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 β* G}
(hf : IsPGroup p f.ker) : β 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 β* G}
(hf : Function.Injective f) : β 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) :
β 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 β* 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 β* 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 β Sylow p G := fun P => Classical.choose (h_exists P)
have hg : β 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 [β hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H β* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H β* 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] :
MulAction Ξ± (Sylow p G) where
smul g P :=
β¨(g β’ 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 (Β· β gβ»ΒΉ β’ Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g β’ s) gβ»ΒΉ Q (hS (smul_mem_pointwise_smul s g P hs))))β©
one_smul P := Sylow.ext (one_smul Ξ± 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 {Ξ± : Type*} [Group Ξ±] [MulDistribMulAction Ξ± G] {g : Ξ±}
{P : Sylow p G} : β(g β’ P) = g β’ (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 β’ P = MulAut.conj g β’ P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
β(g β’ P) = MulAut.conj g β’ (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : β(g β’ P) = MulAut.conj g β’ (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : βP β€ H) (h : H) : β(h β’ P) β€ 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 : βP β€ H) (h : H) :
h β’ P.subtype hP = (h β’ 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 β’ P = P β g β (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, β inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
β¨fun β¨a, b, cβ© => c βΈ by simpa [mul_assoc] using b,
fun hh => β¨(MulAut.conj g)β»ΒΉ h, hh, MulAut.apply_inv_self G (MulAut.conj g) hβ©β©
#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 β’ 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 β fixedPoints H (Sylow p G) β H β€ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, β 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 β (Q : Subgroup G).normalizer = P β 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 β fixedPoints P (Sylow p G) β P β€ Q := by
rw [P.sylow_mem_fixedPoints_iff, β 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) :=
β¨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S β fixedPoints R (orbit G P) β S.1 β fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ β R.1 β€ S := R.2.sylow_mem_fixedPoints_iff
_ β S.1.1 = R := β¨fun h => R.3 S.1.2 h, ge_of_eqβ©
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [β 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)) := ?_
_ β‘ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ β‘ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [β Set.card_singleton (β¨P, mem_orbit_self Pβ© : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact β¨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))β©β©
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) β‘ 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 β fixedPoints P (Sylow p G) β P.1 β€ Q := P.2.sylow_mem_fixedPoints_iff
_ β Q.1 = P.1 := β¨P.3 Q.2, ge_of_eqβ©
_ β Q β {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)] : Β¬p β£ 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 β* (g β’ 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 β* Q := by
rw [β 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 = β€ :=
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 β centralizer (P : Set G))
(hy : gβ»ΒΉ * x * g β centralizer (P : Set G)) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * x * n := by
have h1 : βP β€ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : β(g β’ P) β€ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - β¨z, hz, rflβ©
specialize hy z hz
rwa [β mul_assoc, β eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, β mul_assoc,
eq_inv_mul_iff_mul_eq, β mul_assoc, β mul_assoc] at hy
obtain β¨h, hhβ© :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g β’ P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' β¨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _β©
rw [β 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 β P) (hy : gβ»ΒΉ * x * g β P) :
β n β (P : Subgroup G).normalizer, gβ»ΒΉ * x * g = nβ»ΒΉ * 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 β G β§Έ (P : Subgroup G).normalizer :=
calc
Sylow p G β (β€ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ β orbit G P := by rw [P.orbit_eq_top]
_ β G β§Έ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ β G β§Έ (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 β§Έ (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 β§Έ (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) β£ (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)] : Β¬p β£ (P : Subgroup G).index := by
intro h
letI : Fintype (G β§Έ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain β¨x, hxβ© := exists_prime_orderOf_dvd_card (G := G β§Έ (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 [β zpowers_eq_bot, β Ne, β bot_lt_iff_ne_bot, β
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 βP (P : Subgroup G).normalizer β 0) : Β¬p β£ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [β relindex_mul_index le_normalizer, β card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex β(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := β¨hPβ©
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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((βP : Subgroup N).map N.subtype).normalizer β N = β€ := by
refine' top_le_iff.mp fun g _ => _
obtain β¨n, hnβ© := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) β’ P) P
rw [β inv_mul_cancel_left (βn) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, β mul_smul, β MulAut.conjNormal_val, β 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 (βn * g)).toMonoidHom from
(MulAut.conj (βn * g)).injective)).symm.trans
_
rw [map_map, β 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) β N = G`. -/
theorem Sylow.normalizer_sup_eq_top' {p : β} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p G) (hP : βP β€ N) : (P : Subgroup G).normalizer β N = β€ := by
rw [β 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} {Ξ± : Type v} {Ξ² : 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 β§Έ s)) :
Fintype.card (QuotientGroup.mk β»ΒΉ' t) = Fintype.card s * Fintype.card t := by
rw [β 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 β§Έ H) β MulAction.fixedPoints H (G β§Έ H) β x β normalizer H :=
β¨fun hx =>
have ha : β {y : G β§Έ H}, y β orbit H (x : G β§Έ H) β y = x := mem_fixedPoints'.1 hx _
(inv_mem_iff (G := G)).1
(mem_normalizer_fintype fun n (hn : n β H) =>
have : (nβ»ΒΉ * x)β»ΒΉ * x β H := QuotientGroup.eq.1 (ha β¨β¨nβ»ΒΉ, inv_mem hnβ©, rflβ©)
show _ β H by
rw [mul_inv_rev, inv_inv] at this
convert this
rw [inv_inv]),
fun hx : β n : G, n β H β x * n * xβ»ΒΉ β H =>
mem_fixedPoints'.2 fun y =>
Quotient.inductionOn' y fun y hy =>
QuotientGroup.eq.2
(let β¨β¨b, hbββ©, hbββ© := hy
have hbβ : (b * x)β»ΒΉ * y β H := QuotientGroup.eq.1 hbβ
(inv_mem_iff (G := G)).1 <|
(hx _).2 <|
(mul_mem_cancel_left (inv_mem hbβ)).1 <| by
rw [hx] at hbβ; simpa [mul_inv_rev, mul_assoc] using hbβ)β©
#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 β§Έ H) β
normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H :=
@subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (_) (_)
(MulAction.fixedPoints H (G β§Έ H))
(fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ βΉ_βΊ _).symm)
(by
intros
dsimp only [instHasEquiv]
rw [leftRel_apply (Ξ± := 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 : β} {n : β} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) β‘
card (G β§Έ H) [MOD p] := by
rw [β 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 : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by
have : H.subgroupOf (normalizer H) β 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p β£ card (normalizer H β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ Subgroup.comap ((normalizer H).subtype : normalizer H β* G) H) % p :=
hcard βΈ (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 : β} {n : β} [_hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
p ^ (n + 1) β£ 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 : β} {n : β} [hp : Fact p.Prime]
(hdvd : p ^ (n + 1) β£ card G) {H : Subgroup G} (hH : Fintype.card H = p ^ n) :
β K : Subgroup G, Fintype.card K = p ^ (n + 1) β§ H β€ K :=
let β¨s, hsβ© := exists_eq_mul_left_of_dvd hdvd
have hcard : card (G β§Έ H) = s * p :=
(mul_left_inj' (show card H β 0 from Fintype.card_ne_zero)).1
(by
rw [β 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 β§Έ H.subgroupOf H.normalizer) % p :=
Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H) βΈ
hcard βΈ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
have hm' : p β£ card (normalizer H β§Έ 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 β¨x, hxβ© := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
have hequiv : H β H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
β¨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', β hH, Fintype.card_congr hequiv, β hx, β Fintype.card_zpowers, β
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' β¨β¨y, le_normalizer hyβ©, β¨0, _β©, rflβ©
dsimp only
rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
simpa using hyβ©
#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 β€ m` and `p ^ m` divides the cardinality of `G` -/
theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : β) :
β {n m : β} [_hp : Fact p.Prime] (_hdvd : p ^ m β£ card G) (H : Subgroup G)
(_hH : card H = p ^ n) (_hnm : n β€ m), β K : Subgroup G, card K = p ^ m β§ H β€ 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 β€ m - 1 := le_tsub_of_add_le_right hnm
let β¨K, hKβ© :=
@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) β£ card G := by rwa [tsub_add_cancel_of_le h0m.nat_succ_le]
let β¨K', hK'β© := @exists_subgroup_card_pow_succ _ _ _ _ _ _ hdvd' K hK.1
β¨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2β©)
fun hnm : n = m => β¨H, by simp [hH, hnm]β©
#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 : β) {n : β} [Fact p.Prime]
(hdvd : p ^ n β£ card G) : β K : Subgroup G, Fintype.card K = p ^ n :=
let β¨K, hKβ© := exists_subgroup_card_pow_prime_le p hdvd β₯ (card_bot.trans (by simp)) n.zero_le
β¨K, hK.1β©
#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 : β} (hp : p.Prime) (h : IsPGroup p G)
(hn : p ^ n β€ Nat.card G) : β H : Subgroup G, Nat.card H = p ^ n := by
have : Fact p.Prime := β¨hpβ©
have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
cases nonempty_fintype G
obtain β¨m, hmβ© := h.exists_card_eq
simp_rw [Nat.card_eq_fintype_card] at hm hn β’
refine exists_subgroup_card_pow_prime _ ?_
rw [hm] at hn β’
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 : β} (hp : p.Prime) (h : IsPGroup p G)
{H : Subgroup G} (hn : p ^ n β€ Nat.card H) : β H' β€ H, Nat.card H' = p ^ n := by
obtain β¨H', H'cardβ© := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
refine β¨H'.map H.subtype, map_subtype_le _, ?_β©
rw [β H'card]
let e : H' β* 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 : β} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
(hk : k β€ Nat.card H) (hkβ : k β 0) : β H' β€ H, Nat.card H' β€ k β§ k < p * Nat.card H' := by
obtain β¨m, hmk, hkmβ© : β s, p ^ s β€ k β§ k < p ^ (s + 1) :=
exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hkβ) hp.one_lt
obtain β¨H', H'H, H'cardβ© := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
refine β¨H', H'H, ?_β©
simpa only [pow_succ, H'card] using And.intro hmk hkm
theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p ^ n β£ card G) : p ^ n β£ 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 βΈ 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 : β} [Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : p β£ card P := by
rw [β 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
(card P).Coprime (index (P : Subgroup G)) :=
let β¨_n, hnβ© := IsPGroup.iff_card.mp P.2
hn.symm βΈ (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 : β} [hp : Fact p.Prime] (P : Sylow p G)
(hdvd : p β£ card G) : (P : Subgroup G) β β₯ := by
refine' fun h => hp.out.not_dvd_one _
have key : p β£ 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 : β} [hp : Fact p.Prime] (P : Sylow p G) :
card P = p ^ Nat.factorization (card G) p := by
obtain β¨n, heq : card P = _β© := 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, β hp.out.pow_dvd_iff_dvd_ord_proj (show card G β 0 from card_ne_zero), β 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 : β} [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 β¨P, hHPβ© := (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 βΈ P.3
#align sylow.of_card Sylow.ofCard
@[simp, norm_cast]
theorem coe_ofCard [Fintype G] {p : β} [Fact p.Prime] (H : Subgroup G) [Fintype H]
(card_eq : card H = p ^ (card G).factorization p) : β(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 : β} [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 β¦ ?_ }
obtain β¨x, h1β© := exists_smul_eq G P Q
obtain β¨x, h2β© := exists_smul_eq G P default
rw [Sylow.smul_eq_of_normal] at h1 h2
rw [β h1, β h2]
#align sylow.subsingleton_of_normal Sylow.unique_of_normal
section Pointwise
open Pointwise
theorem characteristic_of_normal {p : β} [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 Ξ¦
show (Ξ¦ β’ 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hn : (βP : Subgroup G).normalizer.Normal) : (βP : Subgroup G).Normal := by
rw [β normalizer_eq_top, β 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 : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
(βP : Subgroup G).normalizer.normalizer = (βP : Subgroup G).normalizer := by
have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
simp_rw [β normalizer_eq_top, Sylow.coe_subtype, β subgroupOf_normalizer_eq le_normalizer, β
subgroupOf_normalizer_eq le_rfl, subgroupOf_self] at this
rw [β subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map,
β 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 : β H : Subgroup G, IsCoatom H β H.Normal) {p : β} [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) : (βP : Subgroup G).Normal :=
normalizer_eq_top.mp
(by
rcases eq_top_or_exists_le_coatom (βP : Subgroup G).normalizer with (heq | β¨K, hK, hNKβ©)
Β· exact heq
Β· haveI := hnc _ hK
have hPK : βP β€ K := le_trans le_normalizer hNK
refine' (hK.1 _).elim
rw [β 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 : β} [Fact p.Prime]
[Finite (Sylow p G)] (P : Sylow p G) : (βP : 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· calc
card (β p : ps, P p) = β p : ps, card (P p) := Fintype.card_pi
_ = β p : ps, p.1 ^ (card G).factorization p.1 := by
congr 1 with β¨p, hpβ©
| exact @card_eq_multiplicity _ _ _ p β¨Nat.prime_of_mem_primeFactors hpβ© (P p) | /-- 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G := by
set ps := (Fintype.card G).primeFactors
-- βTheβ Sylow subgroup for p
let P : β p, Sylow p G := default
have hcomm : Pairwise fun pβ pβ : ps => β x y : G, x β P pβ β y β P pβ β Commute x y := by
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P pβ)) (hn (P pβ))
apply IsPGroup.disjoint_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
refine' MulEquiv.trans (N := β p : ps, P p) _ _
-- There is only one Sylow subgroup for each p, so the inner product is trivial
show (β p : ps, β P : Sylow p G, P) β* β p : ps, P p
Β· -- here we need to help the elaborator with an explicit instantiation
apply @MulEquiv.piCongrRight ps (fun p => β P : Sylow p G, P) (fun p => P p) _ _
rintro β¨p, hpβ©
haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
letI := unique_of_normal _ (hn (P p))
apply MulEquiv.piUnique
show (β p : ps, P p) β* G
apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
apply (bijective_iff_injective_and_card _).mpr
constructor
show Injective _
Β· apply Subgroup.injective_noncommPiCoprod_of_independent
apply independent_of_coprime_order hcomm
rintro β¨pβ, hpββ© β¨pβ, hpββ© hne
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
haveI hpβ' := Fact.mk (Nat.prime_of_mem_primeFactors hpβ)
have hne' : pβ β pβ := by simpa using hne
apply IsPGroup.coprime_card_of_ne pβ pβ hne' _ _ (P pβ).isPGroup' (P pβ).isPGroup'
show card (β p : ps, P p) = card G
Β· calc
card (β p : ps, P p) = β p : ps, card (P p) := Fintype.card_pi
_ = β p : ps, p.1 ^ (card G).factorization p.1 := by
congr 1 with β¨p, hpβ©
| 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 : β {p : β} [Fact p.Prime] (P : Sylow p G), (βP : Subgroup G).Normal) :
(β p : (card G).primeFactors, β P : Sylow p G, (βP : Subgroup G)) β* G | Mathlib_GroupTheory_Sylow |
R : CommRingCat
β’ IsIso (CommRingCat.ofHom (algebraMap (βR) (Localization.Away 1))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
| cases R | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
| Mathlib.Algebra.Category.Ring.Instances.24_0.KsJUUT2FWBN0k2J | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) | Mathlib_Algebra_Category_Ring_Instances |
case mk
Ξ±β : Type ?u.2113
strβ : CommRing Ξ±β
β’ IsIso (CommRingCat.ofHom (algebraMap (β(Bundled.mk Ξ±β)) (Localization.Away 1))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
| exact localization_unit_isIso _ | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
| Mathlib.Algebra.Category.Ring.Instances.24_0.KsJUUT2FWBN0k2J | instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) | Mathlib_Algebra_Category_Ring_Instances |
R : CommRingCat
M : Submonoid βR
β’ Epi (CommRingCat.ofHom (algebraMap (βR) (Localization M))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
theorem IsLocalization.epi {R : Type*} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
β¨fun {T} _ _ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _β©
#align is_localization.epi IsLocalization.epi
instance Localization.epi {R : Type*} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
| rcases R with β¨Ξ±, strβ© | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
| Mathlib.Algebra.Category.Ring.Instances.40_0.KsJUUT2FWBN0k2J | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) | Mathlib_Algebra_Category_Ring_Instances |
case mk
Ξ± : Type ?u.6221
str : CommRing Ξ±
M : Submonoid β(Bundled.mk Ξ±)
β’ Epi (CommRingCat.ofHom (algebraMap (β(Bundled.mk Ξ±)) (Localization M))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
theorem IsLocalization.epi {R : Type*} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
β¨fun {T} _ _ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _β©
#align is_localization.epi IsLocalization.epi
instance Localization.epi {R : Type*} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
rcases R with β¨Ξ±, strβ©
| exact IsLocalization.epi M _ | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
rcases R with β¨Ξ±, strβ©
| Mathlib.Algebra.Category.Ring.Instances.40_0.KsJUUT2FWBN0k2J | instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) | Mathlib_Algebra_Category_Ring_Instances |
R S : CommRingCat
f : R β
S
a : βR
ha : IsUnit (f.hom a)
β’ IsUnit a | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
theorem IsLocalization.epi {R : Type*} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
β¨fun {T} _ _ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _β©
#align is_localization.epi IsLocalization.epi
instance Localization.epi {R : Type*} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
rcases R with β¨Ξ±, strβ©
exact IsLocalization.epi M _
#align localization.epi' Localization.epi'
instance CommRingCat.isLocalRingHom_comp {R S T : CommRingCat} (f : R βΆ S) (g : S βΆ T)
[IsLocalRingHom g] [IsLocalRingHom f] : IsLocalRingHom (f β« g) :=
_root_.isLocalRingHom_comp _ _
set_option linter.uppercaseLean3 false in
#align CommRing.is_local_ring_hom_comp CommRingCat.isLocalRingHom_comp
theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
| convert f.inv.isUnit_map ha | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
| Mathlib.Algebra.Category.Ring.Instances.52_0.KsJUUT2FWBN0k2J | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom | Mathlib_Algebra_Category_Ring_Instances |
case h.e'_3
R S : CommRingCat
f : R β
S
a : βR
ha : IsUnit (f.hom a)
β’ a = f.inv (f.hom a) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import algebra.category.Ring.instances from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
/-!
# Ring-theoretic results in terms of categorical languages
-/
open CategoryTheory
instance localization_unit_isIso (R : CommRingCat) :
IsIso (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) :=
IsIso.of_iso (IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso
#align localization_unit_is_iso localization_unit_isIso
instance localization_unit_isIso' (R : CommRingCat) :
@IsIso CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R (Localization.Away (1 : R))) := by
cases R
exact localization_unit_isIso _
#align localization_unit_is_iso' localization_unit_isIso'
theorem IsLocalization.epi {R : Type*} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
[Algebra R S] [IsLocalization M S] : Epi (CommRingCat.ofHom <| algebraMap R S) :=
β¨fun {T} _ _ => @IsLocalization.ringHom_ext R _ M S _ _ T _ _ _ _β©
#align is_localization.epi IsLocalization.epi
instance Localization.epi {R : Type*} [CommRing R] (M : Submonoid R) :
Epi (CommRingCat.ofHom <| algebraMap R <| Localization M) :=
IsLocalization.epi M _
#align localization.epi Localization.epi
instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
@Epi CommRingCat _ R _ (CommRingCat.ofHom <| algebraMap R <| Localization M : _) := by
rcases R with β¨Ξ±, strβ©
exact IsLocalization.epi M _
#align localization.epi' Localization.epi'
instance CommRingCat.isLocalRingHom_comp {R S T : CommRingCat} (f : R βΆ S) (g : S βΆ T)
[IsLocalRingHom g] [IsLocalRingHom f] : IsLocalRingHom (f β« g) :=
_root_.isLocalRingHom_comp _ _
set_option linter.uppercaseLean3 false in
#align CommRing.is_local_ring_hom_comp CommRingCat.isLocalRingHom_comp
theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
convert f.inv.isUnit_map ha
| exact (RingHom.congr_fun f.hom_inv_id _).symm | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom :=
{ map_nonunit := fun a ha => by
convert f.inv.isUnit_map ha
| Mathlib.Algebra.Category.Ring.Instances.52_0.KsJUUT2FWBN0k2J | theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R β
S) : IsLocalRingHom f.hom | Mathlib_Algebra_Category_Ring_Instances |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
β’ (coevaluation K V) 1 =
let bV := Basis.ofVectorSpace K V;
β i : β(Basis.ofVectorSpaceIndex K V), bV i ββ[K] Basis.coord bV i | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
| simp only [coevaluation, id] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
β’ ((Basis.constr (Basis.singleton Unit K) K) fun x =>
β i : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i)
1 =
β i : β(Basis.ofVectorSpaceIndex K V), (Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
| rw [(Basis.singleton Unit K).constr_apply_fintype K] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
β’ β i : Unit,
(Basis.equivFun (Basis.singleton Unit K)) 1 i β’
β i : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i =
β i : β(Basis.ofVectorSpaceIndex K V), (Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
| simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton] | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
| Mathlib.LinearAlgebra.Coevaluation.48_0.2OSHLJKAlhD35xC | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
β’ LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V) =
β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V)) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
| letI := Classical.decEq (Basis.ofVectorSpaceIndex K V) | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V) =
β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V)) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
| apply TensorProduct.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ LinearMap.comprβ (mk K (Module.Dual K V) K)
(LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V)) =
LinearMap.comprβ (mk K (Module.Dual K V) K)
(β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V))) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
| apply (Basis.ofVectorSpace K V).dualBasis.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ β (i : β(Basis.ofVectorSpaceIndex K V)),
(LinearMap.comprβ (mk K (Module.Dual K V) K)
(LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) i) =
(LinearMap.comprβ (mk K (Module.Dual K V) K)
(β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V))))
((Basis.dualBasis (Basis.ofVectorSpace K V)) i) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; | intro j | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.comprβ (mk K (Module.Dual K V) K)
(LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j) =
(LinearMap.comprβ (mk K (Module.Dual K V) K)
(β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V))))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; | apply LinearMap.ext_ring | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ ((LinearMap.comprβ (mk K (Module.Dual K V) K)
(LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j))
1 =
((LinearMap.comprβ (mk K (Module.Dual K V) K)
(β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V))))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j))
1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
| rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ββ
β(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V))) ββ
LinearMap.lTensor (Module.Dual K V) (coevaluation K V))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] 1) =
(β(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V))) ββ β(TensorProduct.rid K (Module.Dual K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] 1) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
| simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
((LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)))
((LinearMap.lTensor (Module.Dual K V) (coevaluation K V))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] 1))) =
(LinearEquiv.symm (TensorProduct.lid K (Module.Dual K V)))
((TensorProduct.rid K (Module.Dual K V)) ((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] 1)) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
| rw [rid_tmul, one_smul, lid_symm_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
((LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)))
((LinearMap.lTensor (Module.Dual K V) (coevaluation K V))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] 1))) =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
| simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
((LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K]
β i : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i)) =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
| rw [TensorProduct.tmul_sum, map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
(β x : β(Basis.ofVectorSpaceIndex K V),
(LinearEquiv.symm (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)))
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K]
(Basis.ofVectorSpace K V) x ββ[K] Basis.coord (Basis.ofVectorSpace K V) x)) =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; | simp only [assoc_symm_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
(β x : β(Basis.ofVectorSpaceIndex K V),
((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] (Basis.ofVectorSpace K V) x) ββ[K]
Basis.coord (Basis.ofVectorSpace K V) x) =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
| rw [map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ β x : β(Basis.ofVectorSpaceIndex K V),
(LinearMap.rTensor (Module.Dual K V) (contractLeft K V))
(((Basis.dualBasis (Basis.ofVectorSpace K V)) j ββ[K] (Basis.ofVectorSpace K V) x) ββ[K]
Basis.coord (Basis.ofVectorSpace K V) x) =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; | simp only [LinearMap.rTensor_tmul, contractLeft_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ β x : β(Basis.ofVectorSpaceIndex K V),
((Basis.dualBasis (Basis.ofVectorSpace K V)) j) ((Basis.ofVectorSpace K V) x) ββ[K]
Basis.coord (Basis.ofVectorSpace K V) x =
1 ββ[K] (Basis.dualBasis (Basis.ofVectorSpace K V)) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
| simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (β x : β(Basis.ofVectorSpaceIndex K V), if x = j then 1 ββ[K] Basis.coord (Basis.ofVectorSpace K V) x else 0) =
1 ββ[K] Basis.coord (Basis.ofVectorSpace K V) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
| rw [Finset.sum_ite_eq'] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
| Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (if j β Finset.univ then 1 ββ[K] Basis.coord (Basis.ofVectorSpace K V) j else 0) =
1 ββ[K] Basis.coord (Basis.ofVectorSpace K V) j | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; | simp only [Finset.mem_univ, if_true] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; | Mathlib.LinearAlgebra.Coevaluation.60_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
β’ LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V) =
β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
| letI := Classical.decEq (Basis.ofVectorSpaceIndex K V) | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V) =
β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
| apply TensorProduct.ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ LinearMap.comprβ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V)) =
LinearMap.comprβ (mk K K V) (β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V)) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
| apply LinearMap.ext_ring | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.comprβ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V)))
1 =
(LinearMap.comprβ (mk K K V) (β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V))) 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; | apply (Basis.ofVectorSpace K V).ext | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
β’ β (i : β(Basis.ofVectorSpaceIndex K V)),
((LinearMap.comprβ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V)))
1)
((Basis.ofVectorSpace K V) i) =
((LinearMap.comprβ (mk K K V) (β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V))) 1)
((Basis.ofVectorSpace K V) i) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; | intro j | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ ((LinearMap.comprβ (mk K K V)
(LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V)))
1)
((Basis.ofVectorSpace K V) j) =
((LinearMap.comprβ (mk K K V) (β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V))) 1)
((Basis.ofVectorSpace K V) j) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
| rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V) ββ
β(TensorProduct.assoc K V (Module.Dual K V) V) ββ LinearMap.rTensor V (coevaluation K V))
(1 ββ[K] (Basis.ofVectorSpace K V) j) =
(β(LinearEquiv.symm (TensorProduct.rid K V)) ββ β(TensorProduct.lid K V)) (1 ββ[K] (Basis.ofVectorSpace K V) j) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
| simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V))
((TensorProduct.assoc K V (Module.Dual K V) V)
((LinearMap.rTensor V (coevaluation K V)) (1 ββ[K] (Basis.ofVectorSpace K V) j))) =
(LinearEquiv.symm (TensorProduct.rid K V)) ((TensorProduct.lid K V) (1 ββ[K] (Basis.ofVectorSpace K V) j)) | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
| rw [lid_tmul, one_smul, rid_symm_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V))
((TensorProduct.assoc K V (Module.Dual K V) V)
((LinearMap.rTensor V (coevaluation K V)) (1 ββ[K] (Basis.ofVectorSpace K V) j))) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
| simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V))
((TensorProduct.assoc K V (Module.Dual K V) V)
((β i : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) i ββ[K] Basis.coord (Basis.ofVectorSpace K V) i) ββ[K]
(Basis.ofVectorSpace K V) j)) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
| rw [TensorProduct.sum_tmul, map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V))
(β x : β(Basis.ofVectorSpaceIndex K V),
(TensorProduct.assoc K V (Module.Dual K V) V)
(((Basis.ofVectorSpace K V) x ββ[K] Basis.coord (Basis.ofVectorSpace K V) x) ββ[K]
(Basis.ofVectorSpace K V) j)) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; | simp only [assoc_tmul] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (LinearMap.lTensor V (contractLeft K V))
(β x : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) x ββ[K] Basis.coord (Basis.ofVectorSpace K V) x ββ[K] (Basis.ofVectorSpace K V) j) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
| rw [map_sum] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ β x : β(Basis.ofVectorSpaceIndex K V),
(LinearMap.lTensor V (contractLeft K V))
((Basis.ofVectorSpace K V) x ββ[K] Basis.coord (Basis.ofVectorSpace K V) x ββ[K] (Basis.ofVectorSpace K V) j) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; | simp only [LinearMap.lTensor_tmul, contractLeft_apply] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ β x : β(Basis.ofVectorSpaceIndex K V),
(Basis.ofVectorSpace K V) x ββ[K] (Basis.coord (Basis.ofVectorSpace K V) x) ((Basis.ofVectorSpace K V) j) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
| simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (β x : β(Basis.ofVectorSpaceIndex K V), if j = x then (Basis.ofVectorSpace K V) x ββ[K] 1 else 0) =
(Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite]
| rw [Finset.sum_ite_eq] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite]
| Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
case H.h
K : Type u
instβΒ³ : Field K
V : Type v
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
instβ : FiniteDimensional K V
this : DecidableEq β(Basis.ofVectorSpaceIndex K V) := Classical.decEq β(Basis.ofVectorSpaceIndex K V)
j : β(Basis.ofVectorSpaceIndex K V)
β’ (if j β Finset.univ then (Basis.ofVectorSpace K V) j ββ[K] 1 else 0) = (Basis.ofVectorSpace K V) j ββ[K] 1 | /-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field `K` this describes the canonical linear map
from `K` to `V β Dual K V` which corresponds to the identity function on `V`.
## Tags
coevaluation, dual module, tensor product
## Future work
* Prove that this is independent of the choice of basis on `V`.
-/
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct BigOperators
universe u v
variable (K : Type u) [Field K]
variable (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
/-- The coevaluation map is a linear map from a field `K` to a finite dimensional
vector space `V`. -/
def coevaluation : K ββ[K] V β[K] Module.Dual K V :=
let bV := Basis.ofVectorSpace K V
(Basis.singleton Unit K).constr K fun _ =>
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i
#align coevaluation coevaluation
theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
β i : Basis.ofVectorSpaceIndex K V, bV i ββ[K] bV.coord i := by
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
#align coevaluation_apply_one coevaluation_apply_one
open TensorProduct
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ββ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ββ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ββ (TensorProduct.rid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [rid_tmul, one_smul, lid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.lTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.tmul_sum, map_sum]; simp only [assoc_symm_tmul]
rw [map_sum]; simp only [LinearMap.rTensor_tmul, contractLeft_apply]
simp only [Basis.coe_dualBasis, Basis.coord_apply, Basis.repr_self_apply, TensorProduct.ite_tmul]
rw [Finset.sum_ite_eq']; simp only [Finset.mem_univ, if_true]
#align contract_left_assoc_coevaluation contractLeft_assoc_coevaluation
/-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite]
rw [Finset.sum_ite_eq]; | simp only [Finset.mem_univ, if_true] | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap := by
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.comprβ_apply, LinearMap.comprβ_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [LinearEquiv.coe_toLinearMap, LinearMap.rTensor_tmul, coevaluation_apply_one]
rw [TensorProduct.sum_tmul, map_sum]; simp only [assoc_tmul]
rw [map_sum]; simp only [LinearMap.lTensor_tmul, contractLeft_apply]
simp only [Basis.coord_apply, Basis.repr_self_apply, TensorProduct.tmul_ite]
rw [Finset.sum_ite_eq]; | Mathlib.LinearAlgebra.Coevaluation.80_0.2OSHLJKAlhD35xC | /-- This lemma corresponds to one of the coherence laws for duals in rigid categories, see
`CategoryTheory.Monoidal.Rigid`. -/
theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ββ
(TensorProduct.assoc K _ _ _).toLinearMap ββ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ββ (TensorProduct.lid K _).toLinearMap | Mathlib_LinearAlgebra_Coevaluation |
π : Type u_1
instββ΄ : NontriviallyNormedField π
E : Type u_2
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace π E
F : Type u_3
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
β’ HasStrictFDerivAt f ((continuousMultilinearCurryFin1 π E F) (p 1)) x | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
| refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _) | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΄ : NontriviallyNormedField π
E : Type u_2
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace π E
F : Type u_3
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
β’ (fun y => βy - (x, x)β * βy.1 - y.2β) =o[nhds (x, x)] fun x => βx.1 - x.2β | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
| refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ© | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΄ : NontriviallyNormedField π
E : Type u_2
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace π E
F : Type u_3
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
β’ Tendsto (fun y => βy - (x, x)β) (nhds (x, x)) (nhds 0) | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
| refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _ | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΄ : NontriviallyNormedField π
E : Type u_2
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace π E
F : Type u_3
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
h : HasFPowerSeriesAt f p x
β’ βid (x, x) - (x, x)β = 0 | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
| rw [_root_.id, sub_self, norm_zero] | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
| Mathlib.Analysis.Calculus.FDeriv.Analytic.36_0.XLJ3uW4JYwyXQcn | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
β’ HasFPowerSeriesOnBall (_root_.fderiv π f)
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
| suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
β’ HasFPowerSeriesOnBall (_root_.fderiv π f)
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· | apply A.congr | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· | Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
β’ Set.EqOn (fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(_root_.fderiv π f) (EMetric.ball x r) | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
| intro z hz | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
z : E
hz : z β EMetric.ball x r
β’ (fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1)) z =
_root_.fderiv π f z | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
| dsimp | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
z : E
hz : z β EMetric.ball x r
β’ (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1) = _root_.fderiv π f z | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
| rw [β h.fderiv_eq, add_sub_cancel'_right] | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
A :
HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
z : E
hz : z β EMetric.ball x r
β’ ββz - xββ < r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
| simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case A
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
β’ HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
| suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case A
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
B :
HasFPowerSeriesOnBall (fun z => FormalMultilinearSeries.changeOrigin p (z - x) 1)
(FormalMultilinearSeries.changeOriginSeries p 1) x r
β’ HasFPowerSeriesOnBall
(fun z => (continuousMultilinearCurryFin1 π E F) (FormalMultilinearSeries.changeOrigin p (z - x) 1))
(ContinuousLinearMap.compFormalMultilinearSeries
(β(ContinuousLinearEquiv.mk (continuousMultilinearCurryFin1 π E F).toLinearEquiv))
(FormalMultilinearSeries.changeOriginSeries p 1))
x r
case B
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
β’ HasFPowerSeriesOnBall (fun z => FormalMultilinearSeries.changeOrigin p (z - x) 1)
(FormalMultilinearSeries.changeOriginSeries p 1) x r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
| exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case B
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : HasFPowerSeriesOnBall f p x r
β’ HasFPowerSeriesOnBall (fun z => FormalMultilinearSeries.changeOrigin p (z - x) 1)
(FormalMultilinearSeries.changeOriginSeries p 1) x r | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
| simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
| Mathlib.Analysis.Calculus.FDeriv.Analytic.87_0.XLJ3uW4JYwyXQcn | /-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
β’ AnalyticOn π (_root_.fderiv π f) s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
| intro y hy | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
y : E
hy : y β s
β’ AnalyticAt π (_root_.fderiv π f) y | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
| rcases h y hy with β¨p, r, hpβ© | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case intro.intro
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
pβ : FormalMultilinearSeries π E F
rβ : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
y : E
hy : y β s
p : FormalMultilinearSeries π E F
r : ββ₯0β
hp : HasFPowerSeriesOnBall f p y r
β’ AnalyticAt π (_root_.fderiv π f) y | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
| exact hp.fderiv.analyticAt | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
| Mathlib.Analysis.Calculus.FDeriv.Analytic.118_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
β’ AnalyticOn π (_root_.iteratedFDeriv π n f) s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
| induction' n with n IH | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case zero
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
β’ AnalyticOn π (_root_.iteratedFDeriv π Nat.zero f) s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· | rw [iteratedFDeriv_zero_eq_comp] | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case zero
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
β’ AnalyticOn π (β(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 π E F)) β f) s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
| exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case succ
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ AnalyticOn π (_root_.iteratedFDeriv π (Nat.succ n) f) s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· | rw [iteratedFDeriv_succ_eq_comp_left] | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· | Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case succ
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ AnalyticOn π (β(continuousMultilinearCurryLeftEquiv π (fun x => E) F) β _root_.fderiv π (_root_.iteratedFDeriv π n f))
s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
| convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case h.e'_9.h.e'_4
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ β(continuousMultilinearCurryLeftEquiv π (fun x => E) F) = β?g
case g
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ (E βL[π] ContinuousMultilinearMap π (fun i => E) F) βL[π] ContinuousMultilinearMap π (fun x => E) F | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
| case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ (E βL[π] ContinuousMultilinearMap π (fun i => E) F) βL[π] ContinuousMultilinearMap π (fun x => E) F | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
| case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ (E βL[π] ContinuousMultilinearMap π (fun i => E) F) βL[π] ContinuousMultilinearMap π (fun x => E) F | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
| exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F) | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case h.e'_9.h.e'_4
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : β
IH : AnalyticOn π (_root_.iteratedFDeriv π n f) s
β’ β(continuousMultilinearCurryLeftEquiv π (fun x => E) F) =
ββ(ContinuousLinearEquiv.mk (continuousMultilinearCurryLeftEquiv π (fun x => E) F).toLinearEquiv) | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
| rfl | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
| Mathlib.Analysis.Calculus.FDeriv.Analytic.126_0.XLJ3uW4JYwyXQcn | /-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
β’ ContDiffOn π n f s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
| let t := { x | AnalyticAt π f x } | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
β’ ContDiffOn π n f s | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
| suffices : ContDiffOn π n f t | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
this : ContDiffOn π n f t
β’ ContDiffOn π n f s
case this
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
β’ ContDiffOn π n f t | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; | exact this.mono h | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
β’ ContDiffOn π n f t | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
| have H : AnalyticOn π f t := fun x hx => hx | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
H : AnalyticOn π f t
β’ ContDiffOn π n f t | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
| have t_open : IsOpen t := isOpen_analyticAt π f | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
H : AnalyticOn π f t
t_open : IsOpen t
β’ ContDiffOn π n f t | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
have t_open : IsOpen t := isOpen_analyticAt π f
| apply contDiffOn_of_continuousOn_differentiableOn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
have t_open : IsOpen t := isOpen_analyticAt π f
| Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
case this.Hcont
π : Type u_1
instββ΅ : NontriviallyNormedField π
E : Type u_2
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace π E
F : Type u_3
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
p : FormalMultilinearSeries π E F
r : ββ₯0β
f : E β F
x : E
s : Set E
instβ : CompleteSpace F
h : AnalyticOn π f s
n : ββ
t : Set E := {x | AnalyticAt π f x}
H : AnalyticOn π f t
t_open : IsOpen t
β’ β (m : β), βm β€ n β ContinuousOn (fun x => iteratedFDerivWithin π m f t x) t | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
/-!
# Frechet derivatives of analytic functions.
A function expressible as a power series at a point has a Frechet derivative there.
Also the special case in terms of `deriv` when the domain is 1-dimensional.
-/
open Filter Asymptotics
open scoped ENNReal
variable {π : Type*} [NontriviallyNormedField π]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
section fderiv
variable {p : FormalMultilinearSeries π E F} {r : ββ₯0β}
variable {f : E β F} {x : E} {s : Set E}
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by
refine' h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right _)
refine' isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, _, EventuallyEq.rflβ©
refine' (continuous_id.sub continuous_const).norm.tendsto' _ _ _
rw [_root_.id, sub_self, norm_zero]
#align has_fpower_series_at.has_strict_fderiv_at HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x :=
h.hasStrictFDerivAt.hasFDerivAt
#align has_fpower_series_at.has_fderiv_at HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt π f x :=
h.hasFDerivAt.differentiableAt
#align has_fpower_series_at.differentiable_at HasFPowerSeriesAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt π f x β DifferentiableAt π f x
| β¨_, hpβ© => hp.differentiableAt
#align analytic_at.differentiable_at AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt π f x) : DifferentiableWithinAt π f s x :=
h.differentiableAt.differentiableWithinAt
#align analytic_at.differentiable_within_at AnalyticAt.differentiableWithinAt
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
fderiv π f x = continuousMultilinearCurryFin1 π E F (p 1) :=
h.hasFDerivAt.fderiv
#align has_fpower_series_at.fderiv_eq HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
(h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn π f (EMetric.ball x r) := fun _ hy =>
(h.analyticAt_of_mem hy).differentiableWithinAt
#align has_fpower_series_on_ball.differentiable_on HasFPowerSeriesOnBall.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn π f s) : DifferentiableOn π f s := fun y hy =>
(h y hy).differentiableWithinAt
#align analytic_on.differentiable_on AnalyticOn.differentiableOn
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
HasFDerivAt f (continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1)) (x + y) :=
(h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
#align has_fpower_series_on_ball.has_fderiv_at HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
{y : E} (hy : (βyββ : ββ₯0β) < r) :
fderiv π f (x + y) = continuousMultilinearCurryFin1 π E F (p.changeOrigin y 1) :=
(h.hasFDerivAt hy).fderiv
#align has_fpower_series_on_ball.fderiv_eq HasFPowerSeriesOnBall.fderiv_eq
/-- If a function has a power series on a ball, then so does its derivative. -/
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f)
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r := by
suffices A :
HasFPowerSeriesOnBall
(fun z => continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1))
((continuousMultilinearCurryFin1 π E F :
(E[Γ1]βL[π] F) βL[π] E βL[π] F).compFormalMultilinearSeries
(p.changeOriginSeries 1))
x r
Β· apply A.congr
intro z hz
dsimp
rw [β h.fderiv_eq, add_sub_cancel'_right]
simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
suffices B :
HasFPowerSeriesOnBall (fun z => p.changeOrigin (z - x) 1) (p.changeOriginSeries 1) x r
exact
(continuousMultilinearCurryFin1 π E
F).toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall
B
simpa using
((p.hasFPowerSeriesOnBall_changeOrigin 1 (h.r_pos.trans_le h.r_le)).mono h.r_pos
h.r_le).comp_sub
x
#align has_fpower_series_on_ball.fderiv HasFPowerSeriesOnBall.fderiv
/-- If a function is analytic on a set `s`, so is its FrΓ©chet derivative. -/
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
#align analytic_on.fderiv AnalyticOn.fderiv
/-- If a function is analytic on a set `s`, so are its successive FrΓ©chet derivative. -/
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert @ContinuousLinearMap.comp_analyticOn π E
?_ (ContinuousMultilinearMap π (fun _ : Fin (n + 1) β¦ E) F)
?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_
s ?g IH.fderiv
case g =>
exact β(continuousMultilinearCurryLeftEquiv π (fun _ : Fin (n + 1) => E) F)
rfl
#align analytic_on.iterated_fderiv AnalyticOn.iteratedFDeriv
/-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
have t_open : IsOpen t := isOpen_analyticAt π f
apply contDiffOn_of_continuousOn_differentiableOn
Β· | rintro m - | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s := by
let t := { x | AnalyticAt π f x }
suffices : ContDiffOn π n f t; exact this.mono h
have H : AnalyticOn π f t := fun x hx => hx
have t_open : IsOpen t := isOpen_analyticAt π f
apply contDiffOn_of_continuousOn_differentiableOn
Β· | Mathlib.Analysis.Calculus.FDeriv.Analytic.143_0.XLJ3uW4JYwyXQcn | /-- An analytic function is infinitely differentiable. -/
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn π f s) {n : ββ} :
ContDiffOn π n f s | Mathlib_Analysis_Calculus_FDeriv_Analytic |
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