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case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ d (f a) + e (f a) - (f (d a) + f (e a)) = d (f a) - f (d a) + (e (f a) - f (e a)) | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; | ring | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a : A
d e f : Derivation R A A
⊢ ⁅d, e + f⁆ = ⁅d, e⁆ + ⁅d, f⁆ | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by | ext a | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ ⁅d, e + f⁆ a = (⁅d, e⁆ + ⁅d, f⁆) a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; | simp only [commutator_apply, add_apply, map_add] | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ d (e a) + d (f a) - (e (d a) + f (d a)) = d (e a) - e (d a) + (d (f a) - f (d a)) | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; | ring | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a : A
d : Derivation R A A
⊢ ⁅d, d⁆ = 0 | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by | ext a | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d : Derivation R A A
a : A
⊢ ⁅d, d⁆ a = 0 a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; | simp only [commutator_apply, add_apply, map_add] | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d : Derivation R A A
a : A
⊢ d (d a) - d (d a) = 0 a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; | ring_nf | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d : Derivation R A A
a : A
⊢ 0 = 0 a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; | simp | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a : A
d e f : Derivation R A A
⊢ ⁅d, ⁅e, f⁆⁆ = ⁅⁅d, e⁆, f⁆ + ⁅e, ⁅d, f⁆⁆ | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by | ext a | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ ⁅d, ⁅e, f⁆⁆ a = (⁅⁅d, e⁆, f⁆ + ⁅e, ⁅d, f⁆⁆) a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; | simp only [commutator_apply, add_apply, sub_apply, map_sub] | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
d e f : Derivation R A A
a : A
⊢ d (e (f a)) - d (f (e a)) - (e (f (d a)) - f (e (d a))) =
d (e (f a)) - e (d (f a)) - (f (d (e a)) - f (e (d a))) + (e (d (f a)) - e (f (d a)) - (d (f (e a)) - f (d (e a)))) | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; simp only [commutator_apply, add_apply, sub_apply, map_sub]; | ring | instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; simp only [commutator_apply, add_apply, sub_apply, map_sub]; | Mathlib.RingTheory.Derivation.Lie.49_0.lftH2oDc0WOWKWo | instance : LieRing (Derivation R A A) where
add_lie d e f | Mathlib_RingTheory_Derivation_Lie |
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a : A
src✝ : Module R (Derivation R A A) := instModule
r : R
d e : Derivation R A A
⊢ ⁅d, r • e⁆ = r • ⁅d, e⁆ | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; simp only [commutator_apply, add_apply, sub_apply, map_sub]; ring
instance instLieAlgebra : LieAlgebra R (Derivation R A A) :=
{ Derivation.instModule with
lie_smul := fun r d e => by
| ext a | instance instLieAlgebra : LieAlgebra R (Derivation R A A) :=
{ Derivation.instModule with
lie_smul := fun r d e => by
| Mathlib.RingTheory.Derivation.Lie.55_0.lftH2oDc0WOWKWo | instance instLieAlgebra : LieAlgebra R (Derivation R A A) | Mathlib_RingTheory_Derivation_Lie |
case H
R : Type u_1
inst✝² : CommRing R
A : Type u_2
inst✝¹ : CommRing A
inst✝ : Algebra R A
D D1 D2 : Derivation R A A
a✝ : A
src✝ : Module R (Derivation R A A) := instModule
r : R
d e : Derivation R A A
a : A
⊢ ⁅d, r • e⁆ a = (r • ⁅d, e⁆) a | /-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
#align_import ring_theory.derivation.lie from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.
-/
namespace Derivation
variable {R : Type*} [CommRing R]
variable {A : Type*} [CommRing A] [Algebra R A]
variable (D : Derivation R A A) {D1 D2 : Derivation R A A} (a : A)
section LieStructures
/-! # Lie structures -/
/-- The commutator of derivations is again a derivation. -/
instance : Bracket (Derivation R A A) (Derivation R A A) :=
⟨fun D1 D2 =>
mk' ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ fun a b => by
simp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz,
coeFn_coe, LinearMap.sub_apply]
ring⟩
@[simp]
theorem commutator_coe_linear_map : ↑⁅D1, D2⁆ = ⁅(D1 : Module.End R A), (D2 : Module.End R A)⁆ :=
rfl
#align derivation.commutator_coe_linear_map Derivation.commutator_coe_linear_map
theorem commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) :=
rfl
#align derivation.commutator_apply Derivation.commutator_apply
instance : LieRing (Derivation R A A) where
add_lie d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_add d e f := by ext a; simp only [commutator_apply, add_apply, map_add]; ring
lie_self d := by ext a; simp only [commutator_apply, add_apply, map_add]; ring_nf; simp
leibniz_lie d e f := by ext a; simp only [commutator_apply, add_apply, sub_apply, map_sub]; ring
instance instLieAlgebra : LieAlgebra R (Derivation R A A) :=
{ Derivation.instModule with
lie_smul := fun r d e => by
ext a; | simp only [commutator_apply, map_smul, smul_sub, smul_apply] | instance instLieAlgebra : LieAlgebra R (Derivation R A A) :=
{ Derivation.instModule with
lie_smul := fun r d e => by
ext a; | Mathlib.RingTheory.Derivation.Lie.55_0.lftH2oDc0WOWKWo | instance instLieAlgebra : LieAlgebra R (Derivation R A A) | Mathlib_RingTheory_Derivation_Lie |
C : Type u₁
D : Type u₂
E : Type u₃
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
inst✝ : Category.{v₃, u₃} E
F : C × D ⥤ E
W : C
X Y Z : D
f : X ⟶ Y
g : Y ⟶ Z
⊢ F.map (𝟙 W, f ≫ g) = F.map (𝟙 W, f) ≫ F.map (𝟙 W, g) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Lemmas about functors out of product categories.
-/
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
variable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]
@[simp]
theorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :
F.map ((𝟙 X, 𝟙 Y) : (X, Y) ⟶ (X, Y)) = 𝟙 (F.obj (X, Y)) :=
F.map_id (X, Y)
#align category_theory.bifunctor.map_id CategoryTheory.Bifunctor.map_id
@[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by
| rw [← Functor.map_comp, prod_comp, Category.comp_id] | @[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by
| Mathlib.CategoryTheory.Products.Bifunctor.31_0.qAcXEiI0C4dEPVM | @[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) | Mathlib_CategoryTheory_Products_Bifunctor |
C : Type u₁
D : Type u₂
E : Type u₃
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
inst✝ : Category.{v₃, u₃} E
F : C × D ⥤ E
X Y Z : C
W : D
f : X ⟶ Y
g : Y ⟶ Z
⊢ F.map (f ≫ g, 𝟙 W) = F.map (f, 𝟙 W) ≫ F.map (g, 𝟙 W) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Lemmas about functors out of product categories.
-/
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
variable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]
@[simp]
theorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :
F.map ((𝟙 X, 𝟙 Y) : (X, Y) ⟶ (X, Y)) = 𝟙 (F.obj (X, Y)) :=
F.map_id (X, Y)
#align category_theory.bifunctor.map_id CategoryTheory.Bifunctor.map_id
@[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by
rw [← Functor.map_comp, prod_comp, Category.comp_id]
#align category_theory.bifunctor.map_id_comp CategoryTheory.Bifunctor.map_id_comp
@[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by
| rw [← Functor.map_comp, prod_comp, Category.comp_id] | @[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by
| Mathlib.CategoryTheory.Products.Bifunctor.38_0.qAcXEiI0C4dEPVM | @[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) | Mathlib_CategoryTheory_Products_Bifunctor |
C : Type u₁
D : Type u₂
E : Type u₃
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
inst✝ : Category.{v₃, u₃} E
F : C × D ⥤ E
X X' : C
f : X ⟶ X'
Y Y' : D
g : Y ⟶ Y'
⊢ F.map (𝟙 X, g) ≫ F.map (f, 𝟙 Y') = F.map (f, g) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Lemmas about functors out of product categories.
-/
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
variable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]
@[simp]
theorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :
F.map ((𝟙 X, 𝟙 Y) : (X, Y) ⟶ (X, Y)) = 𝟙 (F.obj (X, Y)) :=
F.map_id (X, Y)
#align category_theory.bifunctor.map_id CategoryTheory.Bifunctor.map_id
@[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by
rw [← Functor.map_comp, prod_comp, Category.comp_id]
#align category_theory.bifunctor.map_id_comp CategoryTheory.Bifunctor.map_id_comp
@[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by
rw [← Functor.map_comp, prod_comp, Category.comp_id]
#align category_theory.bifunctor.map_comp_id CategoryTheory.Bifunctor.map_comp_id
@[simp]
theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by
| rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id] | @[simp]
theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by
| Mathlib.CategoryTheory.Products.Bifunctor.45_0.qAcXEiI0C4dEPVM | @[simp]
theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) | Mathlib_CategoryTheory_Products_Bifunctor |
C : Type u₁
D : Type u₂
E : Type u₃
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
inst✝ : Category.{v₃, u₃} E
F : C × D ⥤ E
X X' : C
f : X ⟶ X'
Y Y' : D
g : Y ⟶ Y'
⊢ F.map (f, 𝟙 Y) ≫ F.map (𝟙 X', g) = F.map (f, g) | /-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Lemmas about functors out of product categories.
-/
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
variable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]
@[simp]
theorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :
F.map ((𝟙 X, 𝟙 Y) : (X, Y) ⟶ (X, Y)) = 𝟙 (F.obj (X, Y)) :=
F.map_id (X, Y)
#align category_theory.bifunctor.map_id CategoryTheory.Bifunctor.map_id
@[simp]
theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by
rw [← Functor.map_comp, prod_comp, Category.comp_id]
#align category_theory.bifunctor.map_id_comp CategoryTheory.Bifunctor.map_id_comp
@[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by
rw [← Functor.map_comp, prod_comp, Category.comp_id]
#align category_theory.bifunctor.map_comp_id CategoryTheory.Bifunctor.map_comp_id
@[simp]
theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
#align category_theory.bifunctor.diagonal CategoryTheory.Bifunctor.diagonal
@[simp]
theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by
| rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id] | @[simp]
theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by
| Mathlib.CategoryTheory.Products.Bifunctor.52_0.qAcXEiI0C4dEPVM | @[simp]
theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) | Mathlib_CategoryTheory_Products_Bifunctor |
p : ℕ
G : Type u_1
inst✝ : Group G
P Q : Sylow p G
h : ↑P = ↑Q
⊢ P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by | cases P | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by | Mathlib.GroupTheory.Sylow.74_0.KwMUNfT2GXiDwTx | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q | Mathlib_GroupTheory_Sylow |
case mk
p : ℕ
G : Type u_1
inst✝ : Group G
Q : Sylow p G
toSubgroup✝ : Subgroup G
isPGroup'✝ : IsPGroup p ↥toSubgroup✝
is_maximal'✝ : ∀ {Q : Subgroup G}, IsPGroup p ↥Q → toSubgroup✝ ≤ Q → Q = toSubgroup✝
h : ↑{ toSubgroup := toSubgroup✝, isPGroup' := isPGroup'✝, is_maximal' := is_maximal'✝ } = ↑Q
⊢ { toSubgroup := toSubgroup✝, isPGroup' := isPGroup'✝, is_maximal' := is_maximal'✝ } = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; | cases Q | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; | Mathlib.GroupTheory.Sylow.74_0.KwMUNfT2GXiDwTx | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q | Mathlib_GroupTheory_Sylow |
case mk.mk
p : ℕ
G : Type u_1
inst✝ : Group G
toSubgroup✝¹ : Subgroup G
isPGroup'✝¹ : IsPGroup p ↥toSubgroup✝¹
is_maximal'✝¹ : ∀ {Q : Subgroup G}, IsPGroup p ↥Q → toSubgroup✝¹ ≤ Q → Q = toSubgroup✝¹
toSubgroup✝ : Subgroup G
isPGroup'✝ : IsPGroup p ↥toSubgroup✝
is_maximal'✝ : ∀ {Q : Subgroup G}, IsPGroup p ↥Q → toSubgroup✝ ≤ Q → Q = toSubgroup✝
h :
↑{ toSubgroup := toSubgroup✝¹, isPGroup' := isPGroup'✝¹, is_maximal' := is_maximal'✝¹ } =
↑{ toSubgroup := toSubgroup✝, isPGroup' := isPGroup'✝, is_maximal' := is_maximal'✝ }
⊢ { toSubgroup := toSubgroup✝¹, isPGroup' := isPGroup'✝¹, is_maximal' := is_maximal'✝¹ } =
{ toSubgroup := toSubgroup✝, isPGroup' := isPGroup'✝, is_maximal' := is_maximal'✝ } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; | congr | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; | Mathlib.GroupTheory.Sylow.74_0.KwMUNfT2GXiDwTx | @[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
hϕ : IsPGroup p ↥(MonoidHom.ker ϕ)
h : ↑P ≤ MonoidHom.range ϕ
src✝ : Subgroup K := comap ϕ ↑P
Q : Subgroup K
hQ : IsPGroup p ↥Q
hle : { toSubmonoid := src✝.toSubmonoid, inv_mem' := (_ : ∀ {x : K}, x ∈ src✝.carrier → x⁻¹ ∈ src✝.carrier) } ≤ Q
⊢ Q = { toSubmonoid := src✝.toSubmonoid, inv_mem' := (_ : ∀ {x : K}, x ∈ src✝.carrier → x⁻¹ ∈ src✝.carrier) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
| show Q = P.1.comap ϕ | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
| Mathlib.GroupTheory.Sylow.100_0.KwMUNfT2GXiDwTx | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
hϕ : IsPGroup p ↥(MonoidHom.ker ϕ)
h : ↑P ≤ MonoidHom.range ϕ
src✝ : Subgroup K := comap ϕ ↑P
Q : Subgroup K
hQ : IsPGroup p ↥Q
hle : { toSubmonoid := src✝.toSubmonoid, inv_mem' := (_ : ∀ {x : K}, x ∈ src✝.carrier → x⁻¹ ∈ src✝.carrier) } ≤ Q
⊢ Q = comap ϕ ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
| rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))] | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
| Mathlib.GroupTheory.Sylow.100_0.KwMUNfT2GXiDwTx | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
hϕ : IsPGroup p ↥(MonoidHom.ker ϕ)
h : ↑P ≤ MonoidHom.range ϕ
src✝ : Subgroup K := comap ϕ ↑P
Q : Subgroup K
hQ : IsPGroup p ↥Q
hle : { toSubmonoid := src✝.toSubmonoid, inv_mem' := (_ : ∀ {x : K}, x ∈ src✝.carrier → x⁻¹ ∈ src✝.carrier) } ≤ Q
⊢ Q = comap ϕ (map ϕ Q) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
| exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
| Mathlib.GroupTheory.Sylow.100_0.KwMUNfT2GXiDwTx | /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
h : ↑P ≤ N
⊢ ↑P ≤ MonoidHom.range (Subgroup.subtype N) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by | rwa [subtype_range] | /-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by | Mathlib.GroupTheory.Sylow.127_0.KwMUNfT2GXiDwTx | /-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P✝ : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
P Q : Sylow p G
hP : ↑P ≤ N
hQ : ↑Q ≤ N
h : Sylow.subtype P hP = Sylow.subtype Q hQ
⊢ P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
| rw [SetLike.ext_iff] at h ⊢ | theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
| Mathlib.GroupTheory.Sylow.137_0.KwMUNfT2GXiDwTx | theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
P✝ : Sylow p G
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
P Q : Sylow p G
hP : ↑P ≤ N
hQ : ↑Q ≤ N
h : ∀ (x : ↥N), x ∈ Sylow.subtype P hP ↔ x ∈ Sylow.subtype Q hQ
⊢ ∀ (x : G), x ∈ P ↔ x ∈ Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
| exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩ | theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
| Mathlib.GroupTheory.Sylow.137_0.KwMUNfT2GXiDwTx | theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
c : Set (Subgroup G)
hc1 : c ⊆ {Q | IsPGroup p ↥Q}
hc2 : IsChain (fun x x_1 => x ≤ x_1) c
Q : Subgroup G
hQ : Q ∈ c
x✝ :
↥{
toSubmonoid :=
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) },
inv_mem' :=
(_ :
∀ {g : G},
g ∈
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) }.toSubsemigroup.carrier →
g⁻¹ ∈
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) }.toSubsemigroup.carrier) }
g : G
S : ↑c
hg : g ∈ (fun R => ↑↑R) S
⊢ ∃ k, { val := g, property := (_ : ∃ t ∈ Set.range fun R => ↑↑R, g ∈ t) } ^ p ^ k = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
| refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩) | /-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
| Mathlib.GroupTheory.Sylow.145_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
c : Set (Subgroup G)
hc1 : c ⊆ {Q | IsPGroup p ↥Q}
hc2 : IsChain (fun x x_1 => x ≤ x_1) c
Q : Subgroup G
hQ : Q ∈ c
x✝ :
↥{
toSubmonoid :=
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) },
inv_mem' :=
(_ :
∀ {g : G},
g ∈
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) }.toSubsemigroup.carrier →
g⁻¹ ∈
{
toSubsemigroup :=
{ carrier := ⋃ R, ↑↑R,
mul_mem' := (_ : ∀ {g : G} (h : G), g ∈ ⋃ R, ↑↑R → h ∈ ⋃ R, ↑↑R → g * h ∈ ⋃ R, ↑↑R) },
one_mem' := (_ : ∃ t ∈ Set.range fun R => ↑↑R, 1 ∈ t) }.toSubsemigroup.carrier) }
g : G
S : ↑c
hg : g ∈ (fun R => ↑↑R) S
k : ℕ
hk : { val := g, property := hg } ^ p ^ k = 1
⊢ { val := g, property := (_ : ∃ t ∈ Set.range fun R => ↑↑R, g ∈ t) } ^ p ^ k = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
| rwa [Subtype.ext_iff, coe_pow] at hk ⊢ | /-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
| Mathlib.GroupTheory.Sylow.145_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥(MonoidHom.ker f)
inst✝ : Fintype (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P => exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P => Classical.choose (_ : ∃ Q, comap f ↑Q = ↑P)
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ ↑P = ↑Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by | rw [← hg, h] | /-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by | Mathlib.GroupTheory.Sylow.188_0.KwMUNfT2GXiDwTx | /-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥(MonoidHom.ker f)
inst✝ : Fintype (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P => exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P => Classical.choose (_ : ∃ Q, comap f ↑Q = ↑P)
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ comap f ↑(g Q) = ↑Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; | exact (h_exists Q).choose_spec | /-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; | Mathlib.GroupTheory.Sylow.188_0.KwMUNfT2GXiDwTx | /-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite (Sylow p G)
⊢ Finite (Sylow p ↥H) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
| cases nonempty_fintype (Sylow p G) | /-- 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
| Mathlib.GroupTheory.Sylow.208_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite (Sylow p G)
val✝ : Fintype (Sylow p G)
⊢ Finite (Sylow p ↥H) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
| infer_instance | /-- 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)
| Mathlib.GroupTheory.Sylow.208_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ g • P = P ↔ g ∈ normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
| rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv] | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
| Mathlib.GroupTheory.Sylow.259_0.KwMUNfT2GXiDwTx | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ (∀ (x : G), x ∈ P ↔ x ∈ g • P) ↔ ∀ (h : G), h ∈ ↑P ↔ g⁻¹ * h * g ∈ ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
| exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩ | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
| Mathlib.GroupTheory.Sylow.259_0.KwMUNfT2GXiDwTx | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : G
x✝ : h ∈ g • P
a : G
b : a ∈ ↑↑P
c : ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a = h
⊢ g⁻¹ * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a * g ∈ ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by | simpa [mul_assoc] using b | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by | Mathlib.GroupTheory.Sylow.259_0.KwMUNfT2GXiDwTx | theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : Normal ↑P
⊢ g • P = P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by | simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top] | theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by | Mathlib.GroupTheory.Sylow.270_0.KwMUNfT2GXiDwTx | theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
H : Subgroup G
P : Sylow p G
⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ H ≤ normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
| simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer] | theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
| Mathlib.GroupTheory.Sylow.274_0.KwMUNfT2GXiDwTx | theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
H : Subgroup G
P : Sylow p G
⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ ∀ ⦃x : G⦄, x ∈ H → x • P = P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; | exact Subtype.forall | theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; | Mathlib.GroupTheory.Sylow.274_0.KwMUNfT2GXiDwTx | theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
Q : Sylow p G
⊢ Q ∈ fixedPoints (↥P) (Sylow p G) ↔ P ≤ ↑Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
| rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left] | theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
| Mathlib.GroupTheory.Sylow.288_0.KwMUNfT2GXiDwTx | theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
| classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩ | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
| cases nonempty_fintype (Sylow p G) | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
| have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩ | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
| suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2 | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
| exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2 | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
| rw [← Sylow.ext (H.mp hR)] | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = ↑R | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
| exact R.2 | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
| apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ ¬p ∣ card ↑(orbit G P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
| refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _) | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ 1 ≡ 0 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
| calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ 1 = card ↑(fixedPoints ↥↑P ↑(orbit G P)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
| rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)] | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ card ↑{{ val := P, property := (_ : P ∈ orbit G P) }} = card ↑(fixedPoints ↥↑P ↑(orbit G P)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
| refine' card_congr' (congr_arg _ (Eq.symm _)) | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ fixedPoints ↥↑P ↑(orbit G P) = {{ val := P, property := (_ : P ∈ orbit G P) }} | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
| rw [Set.eq_singleton_iff_unique_mem] | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ { val := P, property := (_ : P ∈ orbit G P) } ∈ fixedPoints ↥↑P ↑(orbit G P) ∧
∀ x ∈ fixedPoints ↥↑P ↑(orbit G P), x = { val := P, property := (_ : P ∈ orbit G P) } | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
| exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩ | /-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
| Mathlib.GroupTheory.Sylow.293_0.KwMUNfT2GXiDwTx | /-- 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) | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
⊢ card (Sylow p G) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
| refine' Sylow.nonempty.elim fun P : Sylow p G => _ | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ card (Sylow p G) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
| have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ card (Sylow p G) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
| have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ Fintype ↑(fixedPoints (↥↑P) (Sylow p G)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
| rw [this] | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ Fintype ↑{P} | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
| infer_instance | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
⊢ card (Sylow p G) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
| have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this] | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
⊢ card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by | simp [this] | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by | Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
this : card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ card (Sylow p G) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
| exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this]) | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
| Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
fin : Fintype ↑(fixedPoints (↥↑P) (Sylow p G))
this : card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ card ↑(fixedPoints (↥↑P) (Sylow p G)) ≡ 1 [MOD p] | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by | rw [this] | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by | Mathlib.GroupTheory.Sylow.322_0.KwMUNfT2GXiDwTx | /-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
| rw [← Classical.choose_spec (exists_smul_eq G P Q)] | /-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
| Mathlib.GroupTheory.Sylow.356_0.KwMUNfT2GXiDwTx | /-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑(Classical.choose (_ : ∃ m, m • P = Q) • P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
| exact P.equivSMul (Classical.choose (exists_smul_eq G P Q)) | /-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
| Mathlib.GroupTheory.Sylow.356_0.KwMUNfT2GXiDwTx | /-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝ : Group G
P : Sylow p G
⊢ stabilizer G P = normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
| ext | theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
| Mathlib.GroupTheory.Sylow.367_0.KwMUNfT2GXiDwTx | theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
case h
p : ℕ
G : Type u_1
inst✝ : Group G
P : Sylow p G
x✝ : G
⊢ x✝ ∈ stabilizer G P ↔ x✝ ∈ normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; | simp [Sylow.smul_eq_iff_mem_normalizer] | theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; | Mathlib.GroupTheory.Sylow.367_0.KwMUNfT2GXiDwTx | theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
| have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le] | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ ↑P ≤ centralizer ↑(zpowers x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by | rwa [le_centralizer_iff, zpowers_le] | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by | Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
| have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ ↑(g • P) ≤ centralizer ↑(zpowers x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
| rw [le_centralizer_iff, zpowers_le] | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ x ∈ centralizer ↑↑(g • P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
| rintro - ⟨z, hz, rfl⟩ | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hz : z ∈ ↑↑P
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
| specialize hy z hz | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hz : z ∈ ↑↑P
hy : z * (g⁻¹ * x * g) = g⁻¹ * x * g * z
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
| rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
| obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1) | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : h • Sylow.subtype (g • P) h2 = Sylow.subtype P h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
| simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : Sylow.subtype ((↑h * g) • P) (_ : ↑((↑h * g) • P) ≤ centralizer ↑(zpowers x)) = Sylow.subtype P h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
| refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩ | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : Sylow.subtype ((↑h * g) • P) (_ : ↑((↑h * g) • P) ≤ centralizer ↑(zpowers x)) = Sylow.subtype P h1
⊢ g⁻¹ * x * g = (↑h * g)⁻¹ * x * (↑h * g) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
| rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right] | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
| Mathlib.GroupTheory.Sylow.372_0.KwMUNfT2GXiDwTx | theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ ↑⊤ ≃ ↑(orbit G P) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by | rw [P.orbit_eq_top] | /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by | Mathlib.GroupTheory.Sylow.396_0.KwMUNfT2GXiDwTx | /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
P : Sylow p G
⊢ G ⧸ stabilizer G P ≃ G ⧸ normalizer ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by | rw [P.stabilizer_eq_normalizer] | /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by | Mathlib.GroupTheory.Sylow.396_0.KwMUNfT2GXiDwTx | /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
⊢ ¬p ∣ index ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
| intro h | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
h : p ∣ index ↑P
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
| letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
h : p ∣ index ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
| rw [index_eq_card (P : Subgroup G)] at h | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h : p ∣ card (G ⧸ ↑P)
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
| obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
| have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm) | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
| let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G)) | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
| have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2 | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
⊢ IsPGroup p ↥Q | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
| apply h.comap_of_ker_isPGroup | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case hϕ
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
⊢ IsPGroup p ↥(MonoidHom.ker (QuotientGroup.mk' ↑P)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
| rw [QuotientGroup.ker_mk'] | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case hϕ
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
⊢ IsPGroup p ↥↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
| exact P.2 | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
hQ : IsPGroup p ↥Q
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
| replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one) | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
hQ : IsPGroup p ↥Q
hp : ¬x = 1
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
| rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
inst✝ : Normal ↑P
fP : FiniteIndex ↑P
this : Fintype (G ⧸ ↑P) := fintypeQuotientOfFiniteIndex ↑P
h✝ : p ∣ card (G ⧸ ↑P)
x : G ⧸ ↑P
hx : orderOf x = p
h : IsPGroup p ↥(zpowers x)
Q : Subgroup G := comap (QuotientGroup.mk' ↑P) (zpowers x)
hQ : IsPGroup p ↥Q
hp : ↑P < comap (QuotientGroup.mk' ↑P) (zpowers x)
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
| exact hp.ne' (P.3 hQ hp.le) | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
| Mathlib.GroupTheory.Sylow.427_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hP : relindex (↑P) (normalizer ↑P) ≠ 0
⊢ ¬p ∣ index ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
| cases nonempty_fintype (Sylow p G) | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hP : relindex (↑P) (normalizer ↑P) ≠ 0
val✝ : Fintype (Sylow p G)
⊢ ¬p ∣ index ↑P | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
| rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer] | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hP : relindex (↑P) (normalizer ↑P) ≠ 0
val✝ : Fintype (Sylow p G)
⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
| haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hP : relindex (↑P) (normalizer ↑P) ≠ 0
val✝ : Fintype (Sylow p G)
this : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
| haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩ | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hP : relindex (↑P) (normalizer ↑P) ≠ 0
val✝ : Fintype (Sylow p G)
this✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
this : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
| replace hP := not_dvd_index_sylow' (P.subtype le_normalizer) | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
val✝ : Fintype (Sylow p G)
this✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
this : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
hP : ¬p ∣ index ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))
⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
| exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G) | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
| Mathlib.GroupTheory.Sylow.446_0.KwMUNfT2GXiDwTx | theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index | Mathlib_GroupTheory_Sylow |
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
⊢ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N = ⊤ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
| refine' top_le_iff.mp fun g _ => _ | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
p✝ : ℕ
G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : Normal N
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
g : G
x✝ : g ∈ ⊤
⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
/-!
# Sylow theorems
The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
* There exists a Sylow `p`-subgroup of `G`.
* All Sylow `p`-subgroups of `G` are conjugate to each other.
* Let `nₚ` be the number of Sylow `p`-subgroups of `G`, then `nₚ` divides the index of the Sylow
`p`-subgroup, `nₚ ≡ 1 [MOD p]`, and `nₚ` is equal to the index of the normalizer of the Sylow
`p`-subgroup in `G`.
## Main definitions
* `Sylow p G` : The type of Sylow `p`-subgroups of `G`.
## Main statements
* `exists_subgroup_card_pow_prime`: A generalization of Sylow's first theorem:
For every prime power `pⁿ` dividing the cardinality of `G`,
there exists a subgroup of `G` of order `pⁿ`.
* `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
Every `p`-subgroup is contained in a Sylow `p`-subgroup.
* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
where `n` is the multiplicity of `p` in the group order.
* `sylow_conjugate`: A generalization of Sylow's second theorem:
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
* `card_sylow_modEq_one`: A generalization of Sylow's third theorem:
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`.
-/
open Fintype MulAction Subgroup
section InfiniteSylow
variable (p : ℕ) (G : Type*) [Group G]
/-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
structure Sylow extends Subgroup G where
isPGroup' : IsPGroup p toSubgroup
is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → toSubgroup ≤ Q → Q = toSubgroup
#align sylow Sylow
variable {p} {G}
namespace Sylow
attribute [coe] Sylow.toSubgroup
--Porting note: Changed to `CoeOut`
instance : CoeOut (Sylow p G) (Subgroup G) :=
⟨Sylow.toSubgroup⟩
-- Porting note: syntactic tautology
-- @[simp]
-- theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
-- rfl
#noalign sylow.to_subgroup_eq_coe
@[ext]
theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P; cases Q; congr
#align sylow.ext Sylow.ext
theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
⟨congr_arg _, ext⟩
#align sylow.ext_iff Sylow.ext_iff
instance : SetLike (Sylow p G) G where
coe := (↑)
coe_injective' _ _ h := ext (SetLike.coe_injective h)
instance : SubgroupClass (Sylow p G) G where
mul_mem := Subgroup.mul_mem _
one_mem _ := Subgroup.one_mem _
inv_mem := Subgroup.inv_mem _
variable (P : Sylow p G)
/-- The action by a Sylow subgroup is the action by the underlying group. -/
instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
inferInstanceAs (MulAction (P : Subgroup G) α)
#align sylow.mul_action_left Sylow.mulActionLeft
variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
/-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
{ P.1.comap ϕ with
isPGroup' := P.2.comap_of_ker_isPGroup ϕ hϕ
is_maximal' := fun {Q} hQ hle => by
show Q = P.1.comap ϕ
rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
#align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
@[simp]
theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
(P.comapOfKerIsPGroup ϕ hϕ h : Subgroup K) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
/-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
P.comapOfKerIsPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
#align sylow.comap_of_injective Sylow.comapOfInjective
@[simp]
theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
↑(P.comapOfInjective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
rfl
#align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
/-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
protected def subtype (h : ↑P ≤ N) : Sylow p N :=
P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [subtype_range])
#align sylow.subtype Sylow.subtype
@[simp]
theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
rfl
#align sylow.coe_subtype Sylow.coe_subtype
theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
#align sylow.subtype_injective Sylow.subtype_injective
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ =>
⟨{ carrier := ⋃ R : c, R
one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
inv_mem' := fun {g} ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
mul_mem' := fun {g} h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
(hc2.total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
P hP)
fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
#align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
instance Sylow.nonempty : Nonempty (Sylow p G) :=
nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
#align sylow.nonempty Sylow.nonempty
noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
Classical.inhabited_of_nonempty Sylow.nonempty
#align sylow.inhabited Sylow.inhabited
theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
(P.2.map f).exists_le_sylow
#align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
(hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
∃ Q : Sylow p G, (Q : Subgroup G).comap H.subtype = P :=
P.exists_comap_eq_of_injective Subtype.coe_injective
#align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
/-- If the kernel of `f : H →* G` is a `p`-group,
then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
(hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
have hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
Fintype.ofInjective g fun P Q h => Sylow.ext (by rw [← hg, h]; exact (h_exists Q).choose_spec)
#align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
/-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
(hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
#align sylow.fintype_of_injective Sylow.fintypeOfInjective
/-- If `H` is a subgroup of `G`, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
Sylow.fintypeOfInjective H.subtype_injective
/-- If `H` is a subgroup of `G`, then `Finite (Sylow p G)` implies `Finite (Sylow p H)`. -/
instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
cases nonempty_fintype (Sylow p G)
infer_instance
open Pointwise
/-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
MulAction α (Sylow p G) where
smul g P :=
⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
inv_smul_eq_iff.mp
(P.3 (hQ.map _) fun s hs =>
(congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
(smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
one_smul P := Sylow.ext (one_smul α P.toSubgroup)
mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
#align sylow.pointwise_mul_action Sylow.pointwiseMulAction
theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
{P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
rfl
#align sylow.pointwise_smul_def Sylow.pointwise_smul_def
instance Sylow.mulAction : MulAction G (Sylow p G) :=
compHom _ MulAut.conj
#align sylow.mul_action Sylow.mulAction
theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
rfl
#align sylow.smul_def Sylow.smul_def
theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
rfl
#align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
rfl
#align sylow.coe_smul Sylow.coe_smul
theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
Subgroup.conj_smul_le_of_le hP h
#align sylow.smul_le Sylow.smul_le
theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
h • P.subtype hP = (h • P).subtype (Sylow.smul_le hP h) :=
Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
#align sylow.smul_subtype Sylow.smul_subtype
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
mem_normalizer_iff, inv_inv]
exact
forall_congr' fun h =>
iff_congr Iff.rfl
⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
#align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
#align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall
#align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
le_antisymm
(le_inf inf_le_left
(sup_eq_right.mp
(Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
(inf_le_inf_left P le_normalizer)
#align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
#align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
/-- A generalization of **Sylow's second theorem**.
If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
⟨fun P Q => by
classical
cases nonempty_fintype (Sylow p G)
have H := fun {R : Sylow p G} {S : orbit G P} =>
calc
S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints R (Sylow p G) :=
forall_congr' fun a => Subtype.ext_iff
_ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
_ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
exact R.2
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
calc
1 = card (fixedPoints P (orbit G P)) := ?_
_ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
_ ≡ 0 [MOD p] := Nat.modEq_zero_iff_dvd.mpr h
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
refine' card_congr' (congr_arg _ (Eq.symm _))
rw [Set.eq_singleton_iff_unique_mem]
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
variable (p) (G)
/-- A generalization of **Sylow's third theorem**.
If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] :
card (Sylow p G) ≡ 1 [MOD p] := by
refine' Sylow.nonempty.elim fun P : Sylow p G => _
have : fixedPoints P.1 (Sylow p G) = {P} :=
Set.ext fun Q : Sylow p G =>
calc
Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
_ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
_ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm
have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by
rw [this]
infer_instance
have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]
exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
#align card_sylow_modeq_one card_sylow_modEq_one
theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
fun h =>
hp.1.ne_one
(Nat.dvd_one.mp
((Nat.modEq_iff_dvd' zero_le_one).mp
((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
#align not_dvd_card_sylow not_dvd_card_sylow
variable {p} {G}
/-- Sylow subgroups are isomorphic -/
nonrec def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
#align sylow.equiv_smul Sylow.equivSMul
/-- Sylow subgroups are isomorphic -/
noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q := by
rw [← Classical.choose_spec (exists_smul_eq G P Q)]
exact P.equivSMul (Classical.choose (exists_smul_eq G P Q))
#align sylow.equiv Sylow.equiv
@[simp]
theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
top_le_iff.mp fun Q _ => exists_smul_eq G P Q
#align sylow.orbit_eq_top Sylow.orbit_eq_top
theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
stabilizer G P = (P : Subgroup G).normalizer := by
ext; simp [Sylow.smul_eq_iff_mem_normalizer]
#align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
(hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
rw [le_centralizer_iff, zpowers_le]
rintro - ⟨z, hz, rfl⟩
specialize hy z hz
rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
obtain ⟨h, hh⟩ :=
exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
#align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
[_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
#align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
/-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
calc
Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
_ ≃ orbit G P := by rw [P.orbit_eq_top]
_ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
_ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
#align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
Fintype (G ⧸ (P : Subgroup G).normalizer) :=
ofEquiv (Sylow p G) P.equivQuotientNormalizer
theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
(P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
card_congr P.equivQuotientNormalizer
#align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) = (P : Subgroup G).normalizer.index :=
(card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
#align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
card (Sylow p G) ∣ (P : Subgroup G).index :=
((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
(index_dvd_of_le le_normalizer)
#align card_sylow_dvd_index card_sylow_dvd_index
theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
[fP : FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index := by
intro h
letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
rw [index_eq_card (P : Subgroup G)] at h
obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
have hQ : IsPGroup p Q := by
apply h.comap_of_ker_isPGroup
rw [QuotientGroup.ker_mk']
exact P.2
replace hp := mt orderOf_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
QuotientGroup.ker_mk'] at hp
exact hp.ne' (P.3 hQ hp.le)
#align not_dvd_index_sylow' not_dvd_index_sylow'
theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index := by
cases nonempty_fintype (Sylow p G)
rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]
haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=
Subgroup.normal_in_normalizer
haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩
replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
#align not_dvd_index_sylow not_dvd_index_sylow
/-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
| obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by
refine' top_le_iff.mp fun g _ => _
| Mathlib.GroupTheory.Sylow.457_0.KwMUNfT2GXiDwTx | /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
of `N`, then `N_G(P) ⊔ N = G`. -/
theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ | Mathlib_GroupTheory_Sylow |
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